<p>I've drawn information from the helpful links below <a href="https://space.stackexchange.com/q/18363/12102">this question</a> and this <a href="https://space.stackexchange.com/a/18365/12102">partial answer</a></p>

<p>I don't think astronomers or planetary scientists expect a geyser of liquid water to be squiring 100 to 200 kilometers above Europa's surface. There are now however at least two published detection of <em>plumes of water vapor</em> that reach that high.</p>

<p>So after doing some reading, asking around, and with helpful conversations in the comments of this questions and the answers here, I think it's pretty clear that there are not geysers on Europa and the term was used incorrectly by the <em>Washington Post</em>. However, Based on the sub-surface jet model, it might not be incorrect to call them jets, as in the <em>BBC</em> article.</p>

<p><a href="https://i.sstatic.net/iGGMZ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/iGGMZ.png" alt="enter image description here"></a> <a href="https://i.sstatic.net/krLJV.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/krLJV.png" alt="enter image description here"></a></p>

<p><strong>above:</strong> NASA hubblesite images from <a href="http://hubblesite.org/newscenter/archive/releases/2016/33/image/b/format/web_print/" rel="nofollow noreferrer">here</a> and <a href="http://hubblesite.org/newscenter/archive/releases/2016/33/image/c/format/web_print/" rel="nofollow noreferrer">here</a> respectively.</p>

<p>The image on the left is reminiscent of a water fountain and probably gives the wrong impression. While there is probably some directionality, it might not be a coherent, tight column like that.</p>

<hr>

<h2>Possibility #1 Squirting Water from an Over-Pressured Ocean:</h2>

<p>While it's easy to imagine that stresses and pressure in the thick ice is compressing the liquid below so much that it would squirt out of a hole like a small hole in a water balloon, probably not. Ice floats. The density of ice is about 0.92 to 0.93 (when very cold) compared to cold water at about 1.00. That means if you dug a hole in the ice and reached the ocean, it would rise, but probably not to the surface. For example, if everything is in equilibrium and the ice were 10km deep, it might end up something like 800 meters below the average surface. There could be short term non-equilibrium, but ice deforms and cracks and refreezes, and over time the system would not build up and maintain huge hydrostatic pressure beyond that caused by the weight of the ice. The mechanics and dynamics of the ice a whole topic in itself.</p>

<p><a href="https://i.sstatic.net/0TDkw.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/0TDkw.jpg" alt="enter image description here"></a></p>

<p><strong>above:</strong> Image of ice fishing from <a href="https://commons.wikimedia.org/wiki/File:Ministry_of_Defence_of_Russia_-_035.jpg" rel="nofollow noreferrer">here</a></p>

<p><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/2/23/Bogstadvannet_IMG_1611.JPG/665px-Bogstadvannet_IMG_1611.JPG" rel="nofollow noreferrer">cf. "I've been waiting all morning - where is the geyser?"</a> </p>

<hr>

<h2>Possibility #2 Squirting Water from an Over-Pressured Trapped Lake:</h2>

<p>This one is a bit easier to imagine. A volume of water trapped within the ice and subjected to huge tectonic forces, a bit like a surface spring of subsurface water on Earth. However a volume of trapped gas would really be helpful here. Since water is nearly incompressible, the loss of even a tiny volume will rapidly reduce the pressure. </p>

<p><a href="https://i.sstatic.net/do63L.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/do63L.png" alt="enter image description here"></a></p>

<p><strong>above:</strong> "Europa's ice-trapped lake sits above the ocean in an illustration, Illustration courtesy Britney Schmidt and Dead Pixel FX, University of Texas at Austin" from National Geographic's <a href="http://news.nationalgeographic.com/news/2011/11/111116-lake-europa-jupiter-moons-earth-space-science-nasa/" rel="nofollow noreferrer">"Great Lakes" Discovered on Jupiter Moon?</a>  (cropped).</p>

<hr>

<h2>Possibility #3 Sub-surface Vent:</h2>

<p><a href="https://i.sstatic.net/27p5z.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/27p5z.png" alt="enter image description here"></a></p>

<p><strong>above:</strong> Figure 3 from Jared James Berg's thesis <a href="https://repositories.lib.utexas.edu/handle/2152/34041" rel="nofollow noreferrer">Simulating water vapor plumes on Europa</a>.</p>

<p><a href="https://repositories.lib.utexas.edu/handle/2152/34041" rel="nofollow noreferrer">Berg's thesis</a> presents <a href="https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo" rel="nofollow noreferrer">Direct Simulation Monte Carlo</a> (DSMC) calculations for a scenario where a reservoir of trapped liquid water below the surface of Europa feeds a <em>volume of water vapor</em> which continuously vents to the surface. If the cross section of the passage has a constriction or narrowing followed by and expansion area, this forms a sort of nozzle which converts some of the random thermal motion of the atoms into a directed flow. The velocity will be somewhat focused upwards, and the magnitude will be of the same order as the thermal velocity.</p>

<p>This model does not rely on high pressures in the crust, but instead "focuses" the thermal motion upwards.</p>

<p>The simulation includes the dynamics small "grains" of water ice in addition to water vapor molecules. At some point early in the trajectory, the mean free path becomes so large that further condensation stops and molecules are traveling on nearly ballistic trajectories under the influence of gravity.</p>

<hr>

<h2>Comments on Geysers and Atmospheres:</h2>

<p>The image of the geyser in the question has initiated some discussion on the effects of atmospheric drag. Geysers on earth reach anywhere from sub 1 meter to as much as 20 meters typically. How much higher would it go if there were no atmosphere?</p>

<p>Here's a calculation (python script below) of isolated droplets launched into various density atmospheres with an initial velocity of 25 m/s just as a ballpark estimate of a tall geyser. In each plot the heigh vs time is calculated for five radii geometrically spaced between 0.1 and 10.0 millimeters. It can be a factor of 10 or 20 effect for the smallest droplets, and maybe a factor of 2 or 3 for the largest, but it doesn't get you to 100 kilometers if there's no air.</p>

<p><a href="https://i.sstatic.net/4kZSl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4kZSl.png" alt="enter image description here"></a></p>

<p>Yes if Europa had Earths atmosphere, the geysers wouldn't be there, but no, there aren't geysers on Europa because there's no atmosphere.</p>

<p>But this kind of calculation is an <em>overestimate</em> of the impact of drag. The effect in a real geyser would probably be much smaller than this kind of calculation suggests.  Take a look at an actual geyser! The mass of the water probably dominates the column, and rapidly transfers momentum to the air in the column by the same drag force. The force ($dp/dt$) is equal and opposite, so actually drag may speed up the air much more than it slows down the water. And of course, this is <em>hot air</em> and steam which are accelerated by buoyancy forces as well.</p>

<p><a href="https://i.sstatic.net/7YZhQ.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7YZhQ.gif" alt="enter image description here"></a></p>

<p>The source is the following video - it really gets going around 02:00, and if you keep watching you can see a <a href="https://www.youtube.com/watch?v=MX0D4oZwCsA" rel="nofollow noreferrer">double rainbow</a> as a bonus! (<a href="https://youtu.be/OQSNhk5ICTI?t=24)" rel="nofollow noreferrer">original</a>)</p>

<p><a href="https://www.youtube.com/watch?v=y8gLhHzPY5M" rel="nofollow noreferrer">https://www.youtube.com/watch?v=y8gLhHzPY5M</a></p>

<pre><code>def Fdrag(x, v):

    if hscale:
        rho = rho0 * np.exp(-x/hscale)
    else:
        rho = rho0

    return -0.5 * rho * v * abs(v) * CD * area

def deriv(X, t):

    x, v     = X
    acc_drag = Fdrag(x, v)/mass

    xdot     = v
    vdot     = -acc_grav + acc_drag

    return np.hstack((xdot, vdot))

class Droplet(object):
    def __init__(self, r):

        self.r = r

class Atmosphere(object):
    def __init__(self, rho0, hscale=None):

        self.rho0   = rho0
        self.hscale = hscale

import numpy as np
from scipy.integrate import odeint as ODEint
import matplotlib.pyplot as plt
import copy

CD_sphere    = 0.47
H2O_density  = 1E+03        # kg/m^3

hscale_Earth = 1E+04        # meters
g_Earth      = 9.8          # m/s^2
rho0_Earth   = 1.3          # kg/m^3

# make some droplets

v0_all = 25.    # m/s

radii = np.logspace(-2, -4, 5)  # meters

droplets = [Droplet(r) for r in radii]    # instantiate

for drop in droplets:     # I love python objects :)
    drop.area   = np.pi * drop.r**2
    drop.volume = (4./3.) * np.pi * drop.r**3
    drop.mass   = H2O_density * drop.volume
    drop.CD     = CD_sphere
    drop.v0     = v0_all
    drop.x0     = 0.0

# make some atmospheres

fractions   = np.array([1E-04, 1E-02, 1])
names       = ['1E-04', '1E-02', '1.0']

atmospheres = []

for frac, name in zip(fractions, names):

    rho0 = rho0_Earth * frac

    atmosphere = Atmosphere(rho0, hscale=hscale_Earth)

    atmosphere.name = name + " of Earth"
    atmosphere.droplets = copy.deepcopy(droplets)  # DEEP copy!
    atmosphere.acc_grav = g_Earth

    atmospheres.append(atmosphere)

tol  = 1E-09
time = np.linspace(0, 6, 200)

xpts = [2.6, 1.8, 1.2, 0.5, 0.3]
ypts = [19, 11.8, 6.3, 3.1, 1.2]
labs = ['0.1', '0.32', '1.0', '3.2', '10'][::-1]
labs = [n + 'mm' for n in labs]

labzip = zip(xpts, ypts, labs)

for atmosphere in atmospheres:

    for drop in atmosphere.droplets:

        X0       = np.hstack((drop.x0, drop.v0))
        mass     = drop.mass
        area     = drop.area
        CD       = drop.CD
        rho0     = atmosphere.rho0
        hscale   = atmosphere.hscale
        acc_grav = atmosphere.acc_grav

        answer, info = ODEint(deriv, X0, time,
                              rtol=tol, atol=tol,
                              full_output=True )
        drop.answer = answer

plt.figure(figsize=[14,5])

for i, atmosphere in enumerate(atmospheres):

    plt.subplot(1,len(atmospheres),i+1)

    for drop in atmosphere.droplets:
        x, v = drop.answer.T
        plt.plot(time, x)

    plt.ylim(0, 34)
    plt.xlim(0, 5.2)

    title = atmosphere.name + ", v0=" + str(v0_all) + " m/s"
    plt.title(title, fontsize=14)

    if i == 2:
        for x, y, s in labzip:
            plt.text(x, y, s, fontsize=12)

    plt.xlabel('time (sec)', fontsize=14)
    plt.ylabel('height (m)', fontsize=14)

plt.suptitle('Single, Isolated droplets in air', fontsize=18)

plt.show()
</code></pre>

CC BY-SA 3.0
2016-09-30T08:38:27.433Z
2018-02-10T17:39:23.857Z
2018-02-10T17:39:23.857Z
uhoh
<p>Technically those aren't "geysers" on Europa, they're <a href="https://en.wikipedia.org/wiki/Cryovolcano">cryovolcanos</a>.    Though that definition may be a bit imprecise as well, but the 100 km eruptions on Europa probably have more in common with volcanic eruptions on Earth than geysers.   Europa undergoes significant tidal flexing which, in combination to it's solid icy crust, could be compared to explosive volcanic eruptions on Earth and not the cyclic geysers, which are far smaller.  </p>

<p>Volcanoes on Earth can shoot material <a href="http://volcano.oregonstate.edu/how-high-can-explosive-eruptions-go-and-how-far-can-debris-and-ash-be-spread">high into the atmosphere</a> on Earth but volcanic ash isn't a good example because it's not so much "shot" into the air as it rides rising hot air to altitudes 30 and 40 KM high.</p>

<p>Magma fountains on Earth can reach a <a href="http://volcano.oregonstate.edu/how-high-can-lava-shoot-air">few hundred meters</a> and <a href="http://www.geo.mtu.edu/volcanoes/hazards/primer/tephra.html">Tephra</a> (anything larger than Ash, from pebbles to large rocks/boulders) can be found as far as a mile from a volcano and can exit the volcano at <a href="https://en.wikipedia.org/wiki/Eruption_column#Formation">several hundred meters per second</a>.   From the Tephra article:</p>

<blockquote>
  <p>Blocks and bombs as large as 8-30 tons have fallen as far away as 1 km
  from their source (Bryant, 1991). Small blocks and bombs have been
  known to travel as far away as 20-80 km (Scott, 1989)! Some of these
  blocks and bombs can have velocities of 75-200 m/s (Bryant, 1991)</p>
</blockquote>

<p>Europa's Tidal-flexing can cause significant pressures within it's icy crust.  See <a href="http://solarsystem.nasa.gov/europa/tidemovie.cfm">here</a>.  and <a href="https://en.wikipedia.org/wiki/Tidal_heating">here</a>. </p>

<p>So, if we draw an Earth-Europa comparison, and assuming my math is right, using 1/2 at^2, all you'd need for material to reach 100 km on Europa is an exit velocity from the ground of a bit over 500 meters per second.   Or, some 20%-30% faster than a 22 caliber bullet.   That's faster than material usually leaves an Earth Volcano, but not all that much faster.   When frozen, Europa's outer icy crust is probably very hard and brittle.  In pressure spots where the pressure exceeds the structural integrity of the icy crust, it's not all that surprising that the results are pretty explosive, leaving the surface at 500 meters per second.  </p>

<p>Ice is very strong and brittle and when cold enough, where the Earth's crust, while rocks are quite brittle and strong, the crust overall can also be quite malleable, especially a few miles down in hotter temperatures under an erupting volcano.  </p>

<p>Very cold ice is resistant to cracking, but under enough pressure, when it finally does crack it can release explosive energy.   Ice cracks in water due to <a href="http://sploid.gizmodo.com/why-do-ice-cubes-crack-when-you-drop-them-in-drinks-1598935375">differential expansion</a>, (Silicate rock doesn't expand with temperature nearly as much as frozen ice does), and while the cracking of an ice cube dropped into water might not seem like much, a big enough ice cube dealing with enough tidal heating can have explosive results.  </p>

<p>While this fun little video might seem unrelated (turn down your volume before watching), it demonstrates the explosive force that rapid warming can have on a hard block of ice.  This isn't steam pressure, as the steam is all escaping.  This is the inside of the block expanding faster than the outside can and the result is impressive.   <a href="https://www.youtube.com/watch?v=epkRd-w3TGw">https://www.youtube.com/watch?v=epkRd-w3TGw</a>   So, with that in mind, explosive energy from a hard, cold 10 mile thick icy surface under the influence of tidal flexing shouldn't be all that surprising.</p>

CC BY-SA 3.0
2016-09-27T16:01:27.763Z
OK - the highest velocity ejecta from Earth volcanos is produced by what? Hydrostatic pressure, or a phase change - dissolved gasses in the magma coming out of 'solution'? I think it's the latter, though I am not a volcanologist. Yes there is pressure, but you need special conditions to turn pressure into efficient directed acceleration. A rifle, or a long tube feeding the volcano. Is this what's happening? Can you find some science that demonstrates that whatever the pressure is, that it can accelerate liquid water to roughly 500 m/sec?
uhoh
CC BY-SA 3.0
2016-09-27T16:13:06.810Z
Because ~500 m/sec happens to be the thermal velocity associated with 293K water molecules, and I am wondering if this cloud is really not a jet nor a geyser nor a cryovolcano, but in fact a big plume of gas who's motion is dominated by the velocity left over from thermal motion. When a cloud of gas expands and cools, it does not loose kinetic energy. It simply becomes ordered motion, not random, and another name for that is "cold". Perhaps "[There is no spoon](https://en.wikipedia.org/wiki/List_of_minor_characters_in_the_Matrix_series#Spoon_Boy_and_the_other_potentials)"!
uhoh
CC BY-SA 3.0
2016-09-27T17:02:49.110Z
Also, the [cryovolcanos](https://en.wikipedia.org/wiki/Cryovolcano) seen on [Enceladus](https://en.wikipedia.org/wiki/Enceladus) for example have the advantage that the surface gravity there is even another factor of ten lower than it is on Europa - almost twelve times lower in fact, at only 0.113 $m/s^2$. I don't think there are cryovolcanos on Europa that can accelerate liquid water directionally up at 500 $m/s$ putting it on a sub-orbital trajectory reaching beyond 100 km altitude. I don't think this - as currently written - answers the question I have asked, when analyzed quantitatively.
uhoh
CC BY-SA 3.0
2016-09-27T17:41:25.073Z
So I'll point out again. The popular and general press is using *geyser* and *jet*, but scan the hard science outlets and you won't see those. Instead you will see *vapor* and *plume*. All of these words have solid scientific definitions, and they are fundamentally different.
uhoh
CC BY-SA 3.0
2016-09-27T21:34:17.800Z
@uhoh  I can delete my answer if it's too far off the mark.   Feel free to add your own.   I'd assumed the pressure from the tidal flexing would be the key factor, but the thermal velocity of (gaseous) water molecules at 293K is a valid point.
userLTK
CC BY-SA 3.0
2016-09-28T02:16:55.917Z
Oh please keep it here! It is a very good answer in the sense that it has a lot of helpful information for people to read! However I think it will turn out to be wrong as written. I'm gently suggesting you add some discussion that there may not in fact be enough pressure to shoot liquid water to 100 to 200 kilometers in this case. You can look at all of the discussion happening [here](http://space.stackexchange.com/q/18363/12102) - one comment links to [this abstract](http://adsabs.harvard.edu/abs/2007GeoRL..34.7202M) and these authors tiptoe http://hubblesite.org/pubinfo/pdf/2016/33/pdf.pdf
uhoh
CC BY-SA 3.0
2016-09-29T11:45:26.187Z
Just BTW regarding the video, it's incredibly sad that someone thought a 25 cent *cloth* ... cloth! ... gardening glove (*the purpose of which is to **keep one's hand clean** when touching **dry soil** - it's not even, for goodness sake, suitable for keeping your hand clean when touching wet soil!!*) would be a good choice for ... wait for it .... handling *molten copper*!  WTF?!
Fattie
CC BY-SA 3.0
2016-09-30T00:02:53.500Z
@uhoh  fair enough, but I think your gas molecule velocity at 20 degrees pretty much debunks my answer, though I'd assumed (assuming may be a mistake) that it was closer to plate tectonics than a geyser.    (and yes, Joe, I agree, the video is by a pretty dumb person, but it does make a good example on how explosive warming and expanding ice can be).
userLTK
CC BY-SA 3.0
2016-09-27T07:40:38.993Z
2016-09-27T07:40:38.993Z
userLTK
<p>I wanted to throw my hat into the mix to flesh out another contributing factor. One reason these "geysers" are so capable of achieving great heights is a lack of atmosphere on Europa to slow them down (and to a lesser extent, weaker gravity). I wrote a basic "physics simulation" in Python 3 which illustrates this purpose, the code for which is below.</p>

<pre><code>import numpy as np

# Define properties of Europa
G = 6.67408E-11     # Gravitational constant, m^3 kg^-1 s^-2
M = 4.7998E22       # Mass of Europa, kg
R = 1560000         # Radius of Europa, m
rho0 = 0.1          # Density of air at surface, kg m^-3
H = 100             # Scale Height of Atmosphere, m

# Define properties of ejected water droplet
C_D = 0.5           # Drag coefficient, unitless
A = np.pi*0.01**2   # Cross-sectional area, m^2
m = 0.001           # Mass of droplet, kg
x = 0               # Initial height above surface, m
v = 500             # Initial velocity, m/s
t = 0               # Initial time, s
dT = 1E-3           # Timestep in "simulation", s

# Run "Simulation"
while v &gt; 0:

    a = -(G*M*m/(R+x)**2 + 0.5*(rho0*np.exp(-x/H))*v**2*C_D*A) / m
    v = v + a*dT
    x = x + v*dT

    t += dT

# Print final results
print('Time:',round(t,4),'(s)\nHeight:',round(x/1000,4),'(km)\nVelocity:',round(v,5),'(m/s)')
</code></pre>

<p>This starts off by defining a droplet with the specified properties (e.g., mass, cross-sectional surface area) and ejects it up with some initial speed from the surface of Europa. As it travels up, the acceleration on it due to gravity and atmospheric drag is constantly calculated and the velocity and subsequently height are updated at every time step. The simulation stops when the droplet reaches the peak of it's upwards journey, indicated by the velocity becoming negative (i.e., the particle has begun falling down again). The final time taken to reach that height, the final height, and the velocity at that height (which should be very nearly zero) are then printed out. The air drag model I used was the basic <a href="https://en.wikipedia.org/wiki/Drag_equation">drag equation</a>. Feel free to substitute something more complex and potentially more realistic. The atmosphere itself, is modeled with a simple exponential model <a href="https://en.wikipedia.org/wiki/Scale_height#Scale_height_used_in_a_simple_atmospheric_pressure_model">as described here</a>.</p>

<p>Now to analyze a few results:</p>

<p><strong>No atmosphere</strong></p>

<p>In this case just set <code>rho0 = 0</code>. This will mean only gravity is acting to slow down the droplet. Without an atmosphere, I get the output:</p>

<pre><code>Time: 413.093 (s)
Height: 101.1158 (km)
Velocity: -0.00046 (m/s)
</code></pre>

<p>To be fair, I chose an ejection velocity which would give me a height of 100 km (which I stole from userLTK's answer). Play around with other ejection velocities as you wish.</p>

<p><strong>Thin Atmosphere</strong></p>

<p>There are two factors here to produce a thin atmosphere. First, setting the density of air at the surface. Without spending the time to dig into more realistic numbers, I'll use <code>rho0 = 0.1</code> ($kg/m^3$). For reference, on Earth it is $1.225\:kg/m^3$ at the surface. The other factor is the scale height. This number controls how rapidly the atmosphere disappears as you go up. A smaller number means a thinner atmosphere that ends more quickly. A larger number means a persistently thicker atmosphere that extends much farther. Again, I didn't look up what a proper scale height should be, but for reference on Earth the scale height is about $8\:km$. Here I used <code>H = 100</code> ($\mathrm{meters}$). In this case, I get the output:</p>

<pre><code>Time: 175.804 (s)
Height: 19.9797 (km)
Velocity: -0.00028 (m/s)
</code></pre>

<p>Even the addition of this thin, weak atmosphere dropped the height achieved by an order of magnitude.</p>

<p><strong>Thick Atmosphere</strong></p>

<p>Let's up the density of the air and make the atmosphere extend more by increasing the scale height. Now <code>rho0 = 0.5</code> ($kg/m^3$) and <code>H = 1000</code> ($\mathrm{meters}$). The output is:</p>

<pre><code>Time: 7.158 (s)
Height: 0.1182 (km)
Velocity: -0.00127 (m/s)
</code></pre>

<p>We've dropped the height by three orders of magnitude compared to the no atmosphere model!</p>

<p><strong>Conclusion</strong></p>

<p>While the physics model is basic, I don't think it is unrealistic. I'm sure with more investigation, more accurate numbers and models could be used, but I believe the results would be more or less the same. On Earth, it is not gravity which primarily limits the height of our geysers, it the strong atmosphere drag. If I run this simulation on Earth, I find a similar water droplet would reach a height of $40\:m$ in our atmosphere, but without that atmosphere, it would get up to $13\:km$, more than 300 times higher. <em>The reason these geysers can reach such large heights is primarily because of the lack of atmosphere drag</em>, compared to what we see on Earth. The weaker gravity does help in the geysers reaching higher heights, but to a lesser extent.</p>

CC BY-SA 3.0
2016-10-24T04:07:47.360Z
Comments are not for extended discussion; this conversation has been [moved to chat](http://chat.stackexchange.com/rooms/47292/discussion-on-answer-by-zephyr-how-can-geysers-on-europa-reach-heights-of-100k).
called2voyage
CC BY-SA 3.0
2016-09-28T14:40:20.850Z
2016-09-28T14:45:29.220Z
2016-09-28T14:45:29.220Z
zephyr
<p>After seeing today's Washington Post's article <a href="https://www.washingtonpost.com/news/speaking-of-science/wp/2016/09/26/nasa-just-saw-europas-geysers-erupting-into-space-again/" rel="noreferrer">NASA just saw Europa’s geysers erupting into space. Again.</a>, and especially the phrase "geysers erupting into space" I checked and Europa has a surface gravity of about 1.3 $m/s^2$, which is substantial - almost the same as the Moon's at about 1.6 $m/s^2$!</p>

<p>The BBC's website article <a href="http://www.bbc.com/news/science-environment-37473617" rel="noreferrer">Europa moon 'spewing water jets'</a> says</p>

<blockquote>
  <p>"Further evidence has been obtained to show that Jupiter's icy moon Europa throws jets of water out into space."</p>
</blockquote>

<p>and </p>

<blockquote>
  <p>"The suggestion is that the jets reach several hundred kilometres in height before then falling back on to Europa."</p>
</blockquote>

<p>On earth, pressure can come from geological hydrostatic forces and by steam produced by geothermal heating, but they rarely rise beyond a few dozen meters. That would be perhaps 100 meters in Europa's gravity.</p>

<p>The images shown of the water above Europa show heights of 100 kilometers and more. </p>

<p>Unlike Earth rock being heavier than water, Europa's ice crust floats. What could be sources of hydrostatic pressure or pressurized steam near the surface that can create such "geysers" or "jets" that can project water to such heights in this gravity?</p>

<p><a href="https://i.sstatic.net/b1Pco.jpg" rel="noreferrer"><img src="https://i.sstatic.net/b1Pco.jpg" alt="enter image description here"></a></p>

<p><strong>above:</strong>  The geyser Strokkur in Iceland, Earth, from <a href="https://en.wikipedia.org/wiki/File:Strokkur,_Iceland.jpg" rel="noreferrer">here</a>.</p>

CC BY-SA 3.0
2016-09-27T04:47:20.827Z
I expect the very thin, almost nonexistent atmosphere is a big factor.
Marc
CC BY-SA 3.0
2016-09-27T04:48:17.567Z
@Marc I expect not. Gravity - it's the law! The liquid water in the image I included would appear roughly the same even if there were no atmosphere. Maybe the use of the term "geyser" is the key? I noticed in the video linked within the article, they definitely use different language.
uhoh
CC BY-SA 3.0
2016-09-27T07:45:18.897Z
As wikipedia says, the surface g=1.3, surface p=1e-12bar,  eruption speed at the surface of Europa can be 500m/s, which is smaller than the speed of the sound?
questionhang
CC BY-SA 3.0
2016-09-27T11:40:42.570Z
Newton's Laws.  Res Ipsa Loquitur  :-)
Carl Witthoft
CC BY-SA 3.0
2016-09-27T17:11:16.060Z
@uhoh Are you sure about that? I believe air resistance should be a non-negligble factor in the height of the geyser. Not saying that's everything, but I'd be willing to bet Europa's lack of an atmosphere plays a noticeable affect it's "geysers" getting higher than they would if an atmosphere did exist. Air resistance scales usually with velocity squared so it can have a substantial effect.
zephyr
CC BY-SA 3.0
2016-09-27T17:25:53.927Z
@zephyr The height of the geyser in that photo is not going to change three or four orders of magnitude if you remove the air. The question is about water that reaches 100 to 200 kilometers. Geysers here are 10 to 20 meters. One power of ten can come from gravity, but there's still three more to deal with. Anything less than a factor of ten is "small" here.
uhoh
CC BY-SA 3.0
2016-09-27T18:49:14.267Z
@uhoh I have to disagree there. To test this, I wrote a super dumb physics simulator which ejected a small $1\:mm$, spherical water droplet in the air, from Europa's surface at $500\:m/s$ (the speed I stole from userLTK). Without air resistance, I get that it reaches a height of $101\:km$ in $\sim7\:min$. If I just throw in a constant atmosphere with a similar density as Earth's (I know, not entirely realistic), I get that the droplet only achieves a height of $1\:km$ in $24\:sec$. This is a factor of 100 difference.
zephyr
CC BY-SA 3.0
2016-09-27T18:53:11.487Z
@uhoh If I try to make it more realistic by scaling the atmospheric density down (as might be expected for Europa) and account for the exponential decay of the atmosphere, it still only reaches a few kilometers. In short, air resistance plays a significant role and geysers on Earth would reach much larger heights (by a factor of 100 by my calculations) were it not for our atmosphere.
zephyr
CC BY-SA 3.0
2016-09-29T11:34:27.933Z
**"The liquid water in the image I included would appear roughly the same even if there were no atmosphere"** that's incredibly incorrect, @uhoh, and seems to be the basis of your question or perhaps misunderstanding.  (Ask anyone who designs things that move through the air .... wings, bullets etc!)
Fattie
CC BY-SA 3.0
2016-09-29T11:40:09.680Z
(... as Zephyr has said.)  Note too that water is pretty much the "worst" thing to push through atmosphere.  It seems that, additionally, the stuff on Europa is more like bullets (small hard chunks).   BTW a fascinating example of the astounding difference atmosphere makes to ballistics, is, the ridiculously, humorously, to us humans highly non-intuitive way that dust behaves on the moon.
Fattie
<water><europa>
How can "Geysers" on Europa reach heights of 100km?
uhoh
# How can "Geysers" on Europa reach heights of 100km?

(Asked by: username_0 on 2016-09-27)

After seeing today's Washington Post's article [NASA just saw Europa’s geysers erupting into space. Again.](https://www.washingtonpost.com/news/speaking-of-science/wp/2016/09/26/nasa-just-saw-europas-geysers-erupting-into-space-again/), and especially the phrase "geysers erupting into space" I checked and Europa has a surface gravity of about 1.3 $m/s^2$, which is substantial - almost the same as the Moon's at about 1.6 $m/s^2$!

The BBC's website article [Europa moon 'spewing water jets'](http://www.bbc.com/news/science-environment-37473617) says

> "Further evidence has been obtained to show that Jupiter's icy moon Europa throws jets of water out into space."

and

> "The suggestion is that the jets reach several hundred kilometres in height before then falling back on to Europa."

On earth, pressure can come from geological hydrostatic forces and by steam produced by geothermal heating, but they rarely rise beyond a few dozen meters. That would be perhaps 100 meters in Europa's gravity.

The images shown of the water above Europa show heights of 100 kilometers and more.

Unlike Earth rock being heavier than water, Europa's ice crust floats. What could be sources of hydrostatic pressure or pressurized steam near the surface that can create such "geysers" or "jets" that can project water to such heights in this gravity?

[![enter image description here](https://i.sstatic.net/b1Pco.jpg)](https://i.sstatic.net/b1Pco.jpg)

**above:** The geyser Strokkur in Iceland, Earth, from [here](https://en.wikipedia.org/wiki/File:Strokkur,_Iceland.jpg).

**Tags:** water, europa
> **Comments:**
> **username_1**: I expect the very thin, almost nonexistent atmosphere is a big factor.  
> **username_0**: @Marc I expect not. Gravity - it's the law! The liquid water in the image I included would appear roughly the same even if there were no atmosphere. Maybe the use of the term "geyser" is the key? I noticed in the video linked within the article, they definitely use different language.  
> **username_2**: As wikipedia says, the surface g=1.3, surface p=1e-12bar,  eruption speed at the surface of Europa can be 500m/s, which is smaller than the speed of the sound?  
> **username_3**: Newton's Laws.  Res Ipsa Loquitur  :-)  
> **username_4**: @uhoh Are you sure about that? I believe air resistance should be a non-negligble factor in the height of the geyser. Not saying that's everything, but I'd be willing to bet Europa's lack of an atmosphere plays a noticeable affect it's "geysers" getting higher than they would if an atmosphere did exist. Air resistance scales usually with velocity squared so it can have a substantial effect.  
> **username_0**: @zephyr The height of the geyser in that photo is not going to change three or four orders of magnitude if you remove the air. The question is about water that reaches 100 to 200 kilometers. Geysers here are 10 to 20 meters. One power of ten can come from gravity, but there's still three more to deal with. Anything less than a factor of ten is "small" here.  
> **username_4**: @uhoh I have to disagree there. To test this, I wrote a super dumb physics simulator which ejected a small $1\:mm$, spherical water droplet in the air, from Europa's surface at $500\:m/s$ (the speed I stole from userLTK). Without air resistance, I get that it reaches a height of $101\:km$ in $\sim7\:min$. If I just throw in a constant atmosphere with a similar density as Earth's (I know, not entirely realistic), I get that the droplet only achieves a height of $1\:km$ in $24\:sec$. This is a factor of 100 difference.  
> **username_4**: @uhoh If I try to make it more realistic by scaling the atmospheric density down (as might be expected for Europa) and account for the exponential decay of the atmosphere, it still only reaches a few kilometers. In short, air resistance plays a significant role and geysers on Earth would reach much larger heights (by a factor of 100 by my calculations) were it not for our atmosphere.  
> **username_5**: **"The liquid water in the image I included would appear roughly the same even if there were no atmosphere"** that's incredibly incorrect, @uhoh, and seems to be the basis of your question or perhaps misunderstanding.  (Ask anyone who designs things that move through the air .... wings, bullets etc!)  
> **username_5**: (... as Zephyr has said.)  Note too that water is pretty much the "worst" thing to push through atmosphere.  It seems that, additionally, the stuff on Europa is more like bullets (small hard chunks).   BTW a fascinating example of the astounding difference atmosphere makes to ballistics, is, the ridiculously, humorously, to us humans highly non-intuitive way that dust behaves on the moon.  

---

## 3 answers

### Answer by username_0 on 2016-09-30. Score: 1 | Accepted answer

I've drawn information from the helpful links below this question and this partial answer
I don't think astronomers or planetary scientists expect a geyser of liquid water to be squiring 100 to 200 kilometers above Europa's surface. There are now however at least two published detection of plumes of water vapor that reach that high.
So after doing some reading, asking around, and with helpful conversations in the comments of this questions and the answers here, I think it's pretty clear that there are not geysers on Europa and the term was used incorrectly by the Washington Post. However, Based on the sub-surface jet model, it might not be incorrect to call them jets, as in the BBC article.
 
above: NASA hubblesite images from here and here respectively.
The image on the left is reminiscent of a water fountain and probably gives the wrong impression. While there is probably some directionality, it might not be a coherent, tight column like that.

Possibility #1 Squirting Water from an Over-Pressured Ocean:
While it's easy to imagine that stresses and pressure in the thick ice is compressing the liquid below so much that it would squirt out of a hole like a small hole in a water balloon, probably not. Ice floats. The density of ice is about 0.92 to 0.93 (when very cold) compared to cold water at about 1.00. That means if you dug a hole in the ice and reached the ocean, it would rise, but probably not to the surface. For example, if everything is in equilibrium and the ice were 10km deep, it might end up something like 800 meters below the average surface. There could be short term non-equilibrium, but ice deforms and cracks and refreezes, and over time the system would not build up and maintain huge hydrostatic pressure beyond that caused by the weight of the ice. The mechanics and dynamics of the ice a whole topic in itself.

above: Image of ice fishing from here
cf. "I've been waiting all morning - where is the geyser?" 

Possibility #2 Squirting Water from an Over-Pressured Trapped Lake:
This one is a bit easier to imagine. A volume of water trapped within the ice and subjected to huge tectonic forces, a bit like a surface spring of subsurface water on Earth. However a volume of trapped gas would really be helpful here. Since water is nearly incompressible, the loss of even a tiny volume will rapidly reduce the pressure. 

above: "Europa's ice-trapped lake sits above the ocean in an illustration, Illustration courtesy Britney Schmidt and Dead Pixel FX, University of Texas at Austin" from National Geographic's "Great Lakes" Discovered on Jupiter Moon?  (cropped).

Possibility #3 Sub-surface Vent:

above: Figure 3 from Jared James Berg's thesis Simulating water vapor plumes on Europa.
Berg's thesis presents Direct Simulation Monte Carlo (DSMC) calculations for a scenario where a reservoir of trapped liquid water below the surface of Europa feeds a volume of water vapor which continuously vents to the surface. If the cross section of the passage has a constriction or narrowing followed by and expansion area, this forms a sort of nozzle which converts some of the random thermal motion of the atoms into a directed flow. The velocity will be somewhat focused upwards, and the magnitude will be of the same order as the thermal velocity.
This model does not rely on high pressures in the crust, but instead "focuses" the thermal motion upwards.
The simulation includes the dynamics small "grains" of water ice in addition to water vapor molecules. At some point early in the trajectory, the mean free path becomes so large that further condensation stops and molecules are traveling on nearly ballistic trajectories under the influence of gravity.

Comments on Geysers and Atmospheres:
The image of the geyser in the question has initiated some discussion on the effects of atmospheric drag. Geysers on earth reach anywhere from sub 1 meter to as much as 20 meters typically. How much higher would it go if there were no atmosphere?
Here's a calculation (python script below) of isolated droplets launched into various density atmospheres with an initial velocity of 25 m/s just as a ballpark estimate of a tall geyser. In each plot the heigh vs time is calculated for five radii geometrically spaced between 0.1 and 10.0 millimeters. It can be a factor of 10 or 20 effect for the smallest droplets, and maybe a factor of 2 or 3 for the largest, but it doesn't get you to 100 kilometers if there's no air.

Yes if Europa had Earths atmosphere, the geysers wouldn't be there, but no, there aren't geysers on Europa because there's no atmosphere.
But this kind of calculation is an overestimate of the impact of drag. The effect in a real geyser would probably be much smaller than this kind of calculation suggests.  Take a look at an actual geyser! The mass of the water probably dominates the column, and rapidly transfers momentum to the air in the column by the same drag force. The force ($dp/dt$) is equal and opposite, so actually drag may speed up the air much more than it slows down the water. And of course, this is hot air and steam which are accelerated by buoyancy forces as well.

The source is the following video - it really gets going around 02:00, and if you keep watching you can see a double rainbow as a bonus! (original)
https://www.youtube.com/watch?v=y8gLhHzPY5M
def Fdrag(x, v):

    if hscale:
        rho = rho0 * np.exp(-x/hscale)
    else:
        rho = rho0

    return -0.5 * rho * v * abs(v) * CD * area

def deriv(X, t):

    x, v     = X
    acc_drag = Fdrag(x, v)/mass

    xdot     = v
    vdot     = -acc_grav + acc_drag

    return np.hstack((xdot, vdot))

class Droplet(object):
    def __init__(self, r):

        self.r = r

class Atmosphere(object):
    def __init__(self, rho0, hscale=None):

        self.rho0   = rho0
        self.hscale = hscale

import numpy as np
from scipy.integrate import odeint as ODEint
import matplotlib.pyplot as plt
import copy

CD_sphere    = 0.47
H2O_density  = 1E+03        # kg/m^3

hscale_Earth = 1E+04        # meters
g_Earth      = 9.8          # m/s^2
rho0_Earth   = 1.3          # kg/m^3

# make some droplets

v0_all = 25.    # m/s

radii = np.logspace(-2, -4, 5)  # meters

droplets = [Droplet(r) for r in radii]    # instantiate

for drop in droplets:     # I love python objects :)
    drop.area   = np.pi * drop.r**2
    drop.volume = (4./3.) * np.pi * drop.r**3
    drop.mass   = H2O_density * drop.volume
    drop.CD     = CD_sphere
    drop.v0     = v0_all
    drop.x0     = 0.0

# make some atmospheres

fractions   = np.array([1E-04, 1E-02, 1])
names       = ['1E-04', '1E-02', '1.0']

atmospheres = []

for frac, name in zip(fractions, names):

    rho0 = rho0_Earth * frac

    atmosphere = Atmosphere(rho0, hscale=hscale_Earth)

    atmosphere.name = name + " of Earth"
    atmosphere.droplets = copy.deepcopy(droplets)  # DEEP copy!
    atmosphere.acc_grav = g_Earth

    atmospheres.append(atmosphere)

tol  = 1E-09
time = np.linspace(0, 6, 200)

xpts = [2.6, 1.8, 1.2, 0.5, 0.3]
ypts = [19, 11.8, 6.3, 3.1, 1.2]
labs = ['0.1', '0.32', '1.0', '3.2', '10'][::-1]
labs = [n + 'mm' for n in labs]

labzip = zip(xpts, ypts, labs)

for atmosphere in atmospheres:

    for drop in atmosphere.droplets:

        X0       = np.hstack((drop.x0, drop.v0))
        mass     = drop.mass
        area     = drop.area
        CD       = drop.CD
        rho0     = atmosphere.rho0
        hscale   = atmosphere.hscale
        acc_grav = atmosphere.acc_grav

        answer, info = ODEint(deriv, X0, time,
                              rtol=tol, atol=tol,
                              full_output=True )
        drop.answer = answer

plt.figure(figsize=[14,5])

for i, atmosphere in enumerate(atmospheres):

    plt.subplot(1,len(atmospheres),i+1)

    for drop in atmosphere.droplets:
        x, v = drop.answer.T
        plt.plot(time, x)

    plt.ylim(0, 34)
    plt.xlim(0, 5.2)

    title = atmosphere.name + ", v0=" + str(v0_all) + " m/s"
    plt.title(title, fontsize=14)

    if i == 2:
        for x, y, s in labzip:
            plt.text(x, y, s, fontsize=12)

    plt.xlabel('time (sec)', fontsize=14)
    plt.ylabel('height (m)', fontsize=14)

plt.suptitle('Single, Isolated droplets in air', fontsize=18)

plt.show()


---

### Answer by username_6 on 2016-09-27. Score: 11

Technically those aren't "geysers" on Europa, they're [cryovolcanos](https://en.wikipedia.org/wiki/Cryovolcano). Though that definition may be a bit imprecise as well, but the 100 km eruptions on Europa probably have more in common with volcanic eruptions on Earth than geysers. Europa undergoes significant tidal flexing which, in combination to it's solid icy crust, could be compared to explosive volcanic eruptions on Earth and not the cyclic geysers, which are far smaller.

Volcanoes on Earth can shoot material [high into the atmosphere](http://volcano.oregonstate.edu/how-high-can-explosive-eruptions-go-and-how-far-can-debris-and-ash-be-spread) on Earth but volcanic ash isn't a good example because it's not so much "shot" into the air as it rides rising hot air to altitudes 30 and 40 KM high.

Magma fountains on Earth can reach a [few hundred meters](http://volcano.oregonstate.edu/how-high-can-lava-shoot-air) and [Tephra](http://www.geo.mtu.edu/volcanoes/hazards/primer/tephra.html) (anything larger than Ash, from pebbles to large rocks/boulders) can be found as far as a mile from a volcano and can exit the volcano at [several hundred meters per second](https://en.wikipedia.org/wiki/Eruption_column#Formation). From the Tephra article:

> Blocks and bombs as large as 8-30 tons have fallen as far away as 1 km
> from their source (Bryant, 1991). Small blocks and bombs have been
> known to travel as far away as 20-80 km (Scott, 1989)! Some of these
> blocks and bombs can have velocities of 75-200 m/s (Bryant, 1991)

Europa's Tidal-flexing can cause significant pressures within it's icy crust. See [here](http://solarsystem.nasa.gov/europa/tidemovie.cfm). and [here](https://en.wikipedia.org/wiki/Tidal_heating).

So, if we draw an Earth-Europa comparison, and assuming my math is right, using 1/2 at^2, all you'd need for material to reach 100 km on Europa is an exit velocity from the ground of a bit over 500 meters per second. Or, some 20%-30% faster than a 22 caliber bullet. That's faster than material usually leaves an Earth Volcano, but not all that much faster. When frozen, Europa's outer icy crust is probably very hard and brittle. In pressure spots where the pressure exceeds the structural integrity of the icy crust, it's not all that surprising that the results are pretty explosive, leaving the surface at 500 meters per second.

Ice is very strong and brittle and when cold enough, where the Earth's crust, while rocks are quite brittle and strong, the crust overall can also be quite malleable, especially a few miles down in hotter temperatures under an erupting volcano.

Very cold ice is resistant to cracking, but under enough pressure, when it finally does crack it can release explosive energy. Ice cracks in water due to [differential expansion](http://sploid.gizmodo.com/why-do-ice-cubes-crack-when-you-drop-them-in-drinks-1598935375), (Silicate rock doesn't expand with temperature nearly as much as frozen ice does), and while the cracking of an ice cube dropped into water might not seem like much, a big enough ice cube dealing with enough tidal heating can have explosive results.

While this fun little video might seem unrelated (turn down your volume before watching), it demonstrates the explosive force that rapid warming can have on a hard block of ice. This isn't steam pressure, as the steam is all escaping. This is the inside of the block expanding faster than the outside can and the result is impressive. <https://www.youtube.com/watch?v=epkRd-w3TGw> So, with that in mind, explosive energy from a hard, cold 10 mile thick icy surface under the influence of tidal flexing shouldn't be all that surprising.

> **Comments:**
> **username_0**: OK - the highest velocity ejecta from Earth volcanos is produced by what? Hydrostatic pressure, or a phase change - dissolved gasses in the magma coming out of 'solution'? I think it's the latter, though I am not a volcanologist. Yes there is pressure, but you need special conditions to turn pressure into efficient directed acceleration. A rifle, or a long tube feeding the volcano. Is this what's happening? Can you find some science that demonstrates that whatever the pressure is, that it can accelerate liquid water to roughly 500 m/sec?  
> **username_0**: Because ~500 m/sec happens to be the thermal velocity associated with 293K water molecules, and I am wondering if this cloud is really not a jet nor a geyser nor a cryovolcano, but in fact a big plume of gas who's motion is dominated by the velocity left over from thermal motion. When a cloud of gas expands and cools, it does not loose kinetic energy. It simply becomes ordered motion, not random, and another name for that is "cold". Perhaps "[There is no spoon](https://en.wikipedia.org/wiki/List_of_minor_characters_in_the_Matrix_series#Spoon_Boy_and_the_other_potentials)"!  
> **username_0**: Also, the [cryovolcanos](https://en.wikipedia.org/wiki/Cryovolcano) seen on [Enceladus](https://en.wikipedia.org/wiki/Enceladus) for example have the advantage that the surface gravity there is even another factor of ten lower than it is on Europa - almost twelve times lower in fact, at only 0.113 $m/s^2$. I don't think there are cryovolcanos on Europa that can accelerate liquid water directionally up at 500 $m/s$ putting it on a sub-orbital trajectory reaching beyond 100 km altitude. I don't think this - as currently written - answers the question I have asked, when analyzed quantitatively.  
> **username_0**: So I'll point out again. The popular and general press is using *geyser* and *jet*, but scan the hard science outlets and you won't see those. Instead you will see *vapor* and *plume*. All of these words have solid scientific definitions, and they are fundamentally different.  
> **username_6**: @uhoh  I can delete my answer if it's too far off the mark.   Feel free to add your own.   I'd assumed the pressure from the tidal flexing would be the key factor, but the thermal velocity of (gaseous) water molecules at 293K is a valid point.  
> **username_0**: Oh please keep it here! It is a very good answer in the sense that it has a lot of helpful information for people to read! However I think it will turn out to be wrong as written. I'm gently suggesting you add some discussion that there may not in fact be enough pressure to shoot liquid water to 100 to 200 kilometers in this case. You can look at all of the discussion happening [here](http://space.stackexchange.com/q/18363/12102) - one comment links to [this abstract](http://adsabs.harvard.edu/abs/2007GeoRL..34.7202M) and these authors tiptoe http://hubblesite.org/pubinfo/pdf/2016/33/pdf.pdf  
> **username_5**: Just BTW regarding the video, it's incredibly sad that someone thought a 25 cent *cloth* ... cloth! ... gardening glove (*the purpose of which is to **keep one's hand clean** when touching **dry soil** - it's not even, for goodness sake, suitable for keeping your hand clean when touching wet soil!!*) would be a good choice for ... wait for it .... handling *molten copper*!  WTF?!  
> **username_6**: @uhoh  fair enough, but I think your gas molecule velocity at 20 degrees pretty much debunks my answer, though I'd assumed (assuming may be a mistake) that it was closer to plate tectonics than a geyser.    (and yes, Joe, I agree, the video is by a pretty dumb person, but it does make a good example on how explosive warming and expanding ice can be).  

---

### Answer by username_4 on 2016-09-28. Score: 7

I wanted to throw my hat into the mix to flesh out another contributing factor. One reason these "geysers" are so capable of achieving great heights is a lack of atmosphere on Europa to slow them down (and to a lesser extent, weaker gravity). I wrote a basic "physics simulation" in Python 3 which illustrates this purpose, the code for which is below.
import numpy as np

# Define properties of Europa
G = 6.67408E-11     # Gravitational constant, m^3 kg^-1 s^-2
M = 4.7998E22       # Mass of Europa, kg
R = 1560000         # Radius of Europa, m
rho0 = 0.1          # Density of air at surface, kg m^-3
H = 100             # Scale Height of Atmosphere, m

# Define properties of ejected water droplet
C_D = 0.5           # Drag coefficient, unitless
A = np.pi*0.01**2   # Cross-sectional area, m^2
m = 0.001           # Mass of droplet, kg
x = 0               # Initial height above surface, m
v = 500             # Initial velocity, m/s
t = 0               # Initial time, s
dT = 1E-3           # Timestep in "simulation", s

# Run "Simulation"
while v > 0:

    a = -(G*M*m/(R+x)**2 + 0.5*(rho0*np.exp(-x/H))*v**2*C_D*A) / m
    v = v + a*dT
    x = x + v*dT

    t += dT

# Print final results
print('Time:',round(t,4),'(s)\nHeight:',round(x/1000,4),'(km)\nVelocity:',round(v,5),'(m/s)')

This starts off by defining a droplet with the specified properties (e.g., mass, cross-sectional surface area) and ejects it up with some initial speed from the surface of Europa. As it travels up, the acceleration on it due to gravity and atmospheric drag is constantly calculated and the velocity and subsequently height are updated at every time step. The simulation stops when the droplet reaches the peak of it's upwards journey, indicated by the velocity becoming negative (i.e., the particle has begun falling down again). The final time taken to reach that height, the final height, and the velocity at that height (which should be very nearly zero) are then printed out. The air drag model I used was the basic drag equation. Feel free to substitute something more complex and potentially more realistic. The atmosphere itself, is modeled with a simple exponential model as described here.
Now to analyze a few results:
No atmosphere
In this case just set rho0 = 0. This will mean only gravity is acting to slow down the droplet. Without an atmosphere, I get the output:
Time: 413.093 (s)
Height: 101.1158 (km)
Velocity: -0.00046 (m/s)

To be fair, I chose an ejection velocity which would give me a height of 100 km (which I stole from userLTK's answer). Play around with other ejection velocities as you wish.
Thin Atmosphere
There are two factors here to produce a thin atmosphere. First, setting the density of air at the surface. Without spending the time to dig into more realistic numbers, I'll use rho0 = 0.1 ($kg/m^3$). For reference, on Earth it is $1.225\:kg/m^3$ at the surface. The other factor is the scale height. This number controls how rapidly the atmosphere disappears as you go up. A smaller number means a thinner atmosphere that ends more quickly. A larger number means a persistently thicker atmosphere that extends much farther. Again, I didn't look up what a proper scale height should be, but for reference on Earth the scale height is about $8\:km$. Here I used H = 100 ($\mathrm{meters}$). In this case, I get the output:
Time: 175.804 (s)
Height: 19.9797 (km)
Velocity: -0.00028 (m/s)

Even the addition of this thin, weak atmosphere dropped the height achieved by an order of magnitude.
Thick Atmosphere
Let's up the density of the air and make the atmosphere extend more by increasing the scale height. Now rho0 = 0.5 ($kg/m^3$) and H = 1000 ($\mathrm{meters}$). The output is:
Time: 7.158 (s)
Height: 0.1182 (km)
Velocity: -0.00127 (m/s)

We've dropped the height by three orders of magnitude compared to the no atmosphere model!
Conclusion
While the physics model is basic, I don't think it is unrealistic. I'm sure with more investigation, more accurate numbers and models could be used, but I believe the results would be more or less the same. On Earth, it is not gravity which primarily limits the height of our geysers, it the strong atmosphere drag. If I run this simulation on Earth, I find a similar water droplet would reach a height of $40\:m$ in our atmosphere, but without that atmosphere, it would get up to $13\:km$, more than 300 times higher. The reason these geysers can reach such large heights is primarily because of the lack of atmosphere drag, compared to what we see on Earth. The weaker gravity does help in the geysers reaching higher heights, but to a lesser extent.

> **Comments:**
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