to Long-Range Artillery - Shelling Paris in WW1
Wikipedia, Space_gun, describes a space gun as a method
of launching an object into space using a large gun or cannon.
Space Guns, provides a description and history of such guns. Of these, the Paris Gun is remarkable as its shells were the first objects man has
sent into space,
(the "Paris Siege Gun").
A large excerpt from that reference follows:
At 7:18 AM on 23 March 1918, a mysterious explosion
occurred in Paris. People thought the explosion was from a bomb dropped by a German
aircraft, but no aircraft had been heard or spotted. 21 more explosions occurred
that day, killing 15 people and wounding 36. Recovery and analysis of munition fragments
indicated that the explosions were caused by artillery shells, fired by a weapon
with extreme range. The best conventional artillery of the time had a range of about
37 kilometers, and the front was about 113 kilometers away.
The Germans had long wanted a way to attack Paris,
in hopes of demoralizing the French public or at least disrupting French war planning.
Aerial bombing was attempted, but the raids proved costly and ineffective. In 1916,
the German Army approached the Krupp manufacturing firm about the possibility of
building an artillery piece that could hit Paris from behind German lines. Krupp's
engineers considered the problem, and had a weapon ready for testing by the spring
The weapon was known as the "Wilhelmgeschuetze (Kaiser
Wilhelm Cannon)", or more generally as the "ParisKanone (Paris Gun)". It was derived
from a 380 millimeter naval gun, into which was inserted a 210 millimeter liner.
A long smooth-bore extension was added, providing a barrel with a total length of
34 meters. The barrel was so long, in fact, that a suspension cable had to be strung
over a superstructure on the top to keep it straight.
The Paris Gun fired a 106 kilogram shell, driven by
an explosive charge of 200 kilograms that produced an acceleration of 7,500 gees
and a muzzle velocity of almost 6,000 kilometers per hour. The gun's maximum range
was 126 kilometers, with the shell reaching a peak altitude of almost 42 kilometers
during its three minutes of flight; the Paris Gun's shells were the first objects
ever sent by humans out of the Earth's atmosphere and into space. The large powder
charge melted the lining of the gun slightly every time it was fired. This meant
that the shells had to be built in a numbered series, in a sequence of increasing
diameters, to be fired in that order until the barrel was replaced and the cycle
began again. The barrels were swapped out on a monthly basis.
Apparently a total of seven Paris Guns were built,
with three put into action. The Germans obtained targeting information from spies
in Paris, who relayed information through Switzerland on where the shells struck
the city. The Paris Guns fired 350 rounds from March through August 1918, killing
256 people and wounding over 600. The Allies tried desperately to destroy the guns
with artillery and aircraft, but the Germans camouflaged the guns and protected
them in concrete emplacements when they were not in use. After the Armistice, the
Allies tried to seize the big guns, but the Germans managed to spirit them away.
To this day nobody is quite sure what happened to them.
Modeling the Path of the Paris Gun Shells
The excerpt contains many scraps of information. The front was about 113 km distant
from Paris. Perhaps as many as 25 shells with different diameters were used. Each
with a weight of 106 kilos?
The muzzle velocity was a bit less than 1667 metres per second. The maximum
range was about 126 km. The flight time was about 3 minutes. The maximum altitude
reached was about 42 km.
Would it be possible to create a spreadsheet model that provided results that were
in the "ball park" of those given in the reference?
Earth is Not Flat
Over short distances, altitude and hence g and atmospheric density, can be given
a horizontal straight-line representation and rectangular coordinates can be employed.
Over longer distances, polar coordinates must be employed to reflect the earth's
curvature. With curvature, velocity along a path is represented by its tangential
and radial components with gravity and buoyancy affecting only the radial component. (Atmospheric resistance affects both components.)
The 3D spreadsheet developed
in Chapter 7 contains the Paris sheet, a sheet that is well suited to modeling sub-orbital and orbital trajectories in atmosphere on a spherical rotating Earth.
Assumptions and Parameters
The centre of a large Paris traffic circle, Place de la Bastille,
was chosen as
the target. From Google Earth its coordinates are Latitude 48.853123 degrees,
Longitude 2.36916 degrees. It is seen next.
Recall that the spreadsheet allows for both a Platform Altitude and a Target Altitude.
Google Earth, the source of the foregoing image, provided an altitude of 41 metres
for the target.
Three Paris Guns were
in action, each about 113 km distant from Paris. As
the Paris guns were
rail mounted, we chose a spot on a railroad track in Tergnier ~ 113 km distant. See the launch site next.
Google Earth gives the launch site altitude as 49 metres and its coordinates
as Lat. 49.6557 degrees, Lon. 3.32924 degrees.
The next image shows both the launch site and the target.
Muzzle velocity was "almost" 6,000 kilometres per hour.
We chose 1,666 metres per second.
Shells were nominally 105 millimetre in radius and varied, throughout a month, after
each round was fired. We picked the 105-millimetre value.
Our spreadsheet entry to represent the weight of the shell is density. Our entry
of 21860 for density results in a sea level weight of 106.00 kg for the shell.
Shells changed in weight with each shot.
Only three spreadsheet input parameters remain to be chosen: the step size
for the calculation, the drag coefficient, and the elevation
angle for maximum range and altitude. A step size of 0.02 seconds was selected. The
drag coefficient can then be estimated by choosing a value to best match the
altitude and range.
A compromise match is made. At an elevation angle of 49 degrees a range
of 130.3 km is achieved with a drag coefficient of .18.
The altitude attained during the flight was 40.67 km.
The flight was said to take about three minutes. How far will Paris move, due to
the rotation of the earth, during flight?
Presuming that the cannon is North East of Paris, it will be moving eastward at a slower
speed than is Paris. A shell that is fired will have the eastward velocity
of the gun and, as it travels southerly, it could be seen to be traveling
slowly than the ground beneath it. This makes the shell appear to have a westward
deflection that the gunner should take into account. There is no force causing
this apparent deflection. It is called the
Coriolis effect. It is an effect
of earth's rotation.
More Than One Way to Aim a Cannon?
Using their spotters, the gunners were able to converge on appropriate azimuth and
elevation values for their shots.
The target range is ~113.221 km. There are two possible elevation
angles that can be used, a more vertical lobbing shot and a lower elevation more direct shot. However, a description of the Paris Gun suggests that its maximum elevation capability was
55 degrees, not sufficiently high in this situation for a lobbing shot.
The parameter table for the low angle shot, in two sections, follows:
The user, gunner, iterates on the Elevation and Azimuth in the left section of the table
until satisfied with the value for Least P-T Range, (Projectile to Target Range),
in the right section, 0.21 metres.
It is like having a spotter to relay the error back to the gunner. If the shell
overshoots the target by too much the gunner lowers the elevation angle. In
practice it would only be extreme chance that would result in a shell falling as close
to the target as is shown foregoing.
A city map of the trajectory and a graph of the altitude are shown next:
The Paris spreadsheet provides the ability to turn off Earth rotation. Without
rotation there is no Coriolis effect. Using the same elevation and azimuth as was
used with earth rotation; the shell misses the target by nearly a km.
A database is developed for modeling travel to the International Space Station.
The Splashdown into the Indian Ocean of the Shuttle's External Tank is modeled.
The thought experiment of Newton that suggests that a cannon ball be shot from a
mountaintop that is high enough to avoid the effects of atmosphere is considered.
Gerald Bull's ambition
of launching a satellite from a cannon is considered.