July 21st Shellings Of Donetsk

Donetsk was shelled on the 19th and 21st of July 2015. There are two cases of special interest here. In the one case an armour piercing shell struck an apartment building and in another case explosive shells struck a hospital. Now let us try to determine where these shells were fired from.

The armour piercing shell

Lugansk-news.com made an interesting article in which they present a video taken at about the same time as the shell struck the apartment building and in the video a shell with a burning tracer can be seen flying at 1:39. The video was taken about 1.3 km north-west of the impact site (google maps location).

In this video we can see that the shell flew by near the 13th floor of the building in the background (verified via a street view maps service, see image below). Since residential buildings have a height of approximately 3.1 meters per floor and the first floor appears to be a bit higher than the other floors, we can deduce that the shells height was about 43m. Furthermore, it can be seen that the shell had a very low slope, it was almost travelling horizontally.

donetsk-arShell-analysis2.jpg

As for the impact site, Lugansk-News reports that it hit at the 7th floor (approximate height 7*3.1=22m). The OSCE report, in addition to confirming the shell type, has confirmed that the shell went through 2 walls before coming to a stop (so it had considerable horizontal velocity at impact). One thing to keep in mind is that 125mm armour-piercing shells dont really have a range greater than 5km (but this refers to the usable range, longer ranges are possible if you don't plan on defeating an armoured target and dont mind trying to hit it from the top).

Now we have two data points which need to match the shells trajectory.
direction-distance.jpg

We can see that the airport begins from about 3km away and the government controlled village of Optyne is about 6 km away.

Trajectory analysis

Now to begin analysing the shells trajectory, we need some equations. Shells trajectories are fundamentally determined by ballistic motion equations combined with aerodynamics, this site provides a spreadsheet for plotting shell trajectories but to use it we need to determine the air resistance parameter, b, which applies a drag force to the shell that is proportional to the speed of the shell.

Fdrag=b*v2

Here, v, represents the velocity. In turn, b, is composed of other terms as follows:

b=0.5*r*S*Cd

Here, r, S and Cd are the air density, cross sectional area of the shell and the aero dynamic drag, respectively. Standard air density is 1.204 kg/m3, 125mm armour piercing shells generally look like this and after the sabot blows off, the radius of the penetrator is 18mm. The hardest parameter to find is Cd, it is a dimensionless quantity that is determined by the shape of the shell in question, fortunately there is a US army report that studies the aerodynamic properties of dart shaped projectiles, using this report with a Mach number of 5 (because these kinds of shells are known to have a launch velocity of about 1800 m/sec) we can find that a Cd value of 0.3-0.5 is appropriate, lets take 0.5 to be safe because the penetrators used in 125mm shells have slightly steeper inclined heads than the one studied in the US army report. Now we can determine the, b, parameter as

b = 0.018×0.018×3.14×1.2041×0.5×0.5 = 0.0003

Now we can finally plug all these values into the spreadsheet and plot some trajectories. As a sanity check, here is a trajectory for firing at 45 degrees:

45deg-calibration.png

as we can see the maximum range is about 8500m (for comparison this site states that a training round which imitates the performance of a 125mm armour piercing shell has a max range under 8km) but this is not a useful range because the shell looses most of its kinetic energy and comes down in an almost vertical fashion (making it hard to aim), thus the usable range is often stated as 4000-5000m.

Ok now that we have verified that the shell trajectory model is roughly correct, lets try to find a firing angle that matches the two observed points. Firing at about 1.4 degrees gives the following result (the spreadsheet that was used to plot this trajectory is available for download at the bottom of the page):

militantsShellDonetsk.png

In this scenario the shell passes the camera at a height of about 43m and then strikes the impact location at about 22m, this scenario suggests that the shell was fired from the militant controlled airport and matches the observed data points.

Next lets consider the case where the shell was fired at 3.1 degrees:

militantsShellDonetsk-3-1deg-scenario.png

Here we can see that the shell could reach the correct height at the location of the camera but then failed to reach the impact site, this scenario suggests that the shell was fired from the government controlled village of Optyne but does not match the observed data points.

Finally lets consider the case where the shell was fired from an angle of 6 degrees:

militantsShellDonetsk-6deg-scenario.png

Here we can see that the shell hit the impact site at the correct location but it did not match the correct height at the camera location. Additionally, in this case the shell had more vertical than horizontal velocity at the end so it would be more likely to knock though a couple floors than a couple walls, yet the OSCE reported that the shell went through a couple walls. This scenario also suggests that the shell was fired from the government controlled village of Optyne but does not match the observed data points.

Measured shell velocity agrees with trajectory model

ban_bang analyzed the shells velocity by studying the distance it travelled over 7 frames and deducing the time per frame as 33.3 msec ( 30 frames per second, 1frame/30fps = 33.3msec ), thus the total time should be 0.233 seconds. However, I believe ban_bang overestimated the distance, using the image below and we can see that the visual features marked by the blue square and green circles match up, this leads us to conclude that the shell travelled 180 meters in 0.233 seconds, thus its velocity was about 770 m/sec:
velocity_analysis.jpg

Now lets compare this measured velocity to the velocities predicted by the trajectory model:

Trajectory Model firing angle [deg] Model Velocity at camera location [m/sec]
1.4 720
3.1 289
6.0 286

As can be seen the measured velocity of 770 m/sec roughly matches the 720 m/sec of the 1.4 degrees launch scenario while the other firing angles completely fail to match the measured velocity.

Tracer burn time reinforces airport launch scenario

Furthermore, remember how the tracer was clearly visible in the video? Lugansk-news suggests that the burn time of the tracer on these kinds of shells is 2-3 seconds (which also corresponds to the shells useful firing range), yet the Optyne launch scenarios require the tracer to burn nearly 3 times as long but the airport launch scenario is consistent with the 2-3 second limit.

Conclusion

Thus the scenarios can be summed up as follows:

scenarioComparison.png

From this we can see that the most consistent launch locations is the militant controlled airport and indeed this is also consistent with the 4000-5000m of useful range which these kinds of shells are usually rated for.
The interested reader can repeat these calculations by downloading my modified spread sheet here, july21-trajectory-analysis.xls (based on this original spreadsheet).

The explosive shells

On this OSCE reports:

On 19 July the SMM observed the aftermath of shelling overnight and conducted crater analyzes in “DPR”-controlled Donetsk city, visiting a total of 12 impact sites. Around 80 Universytetska Street and 69 Shchorsa Street (2.5km north-west of Donetsk city centre), the SMM observed three fresh craters that it concluded had been caused by MBT fragmentation shells (125mm) fired from the north-west. At Hospital No. 23 at 46 Tselinogradska Street (4km west of Donetsk city centre), the SMM observed three fresh craters that it also concluded had been caused by MBT fragmentation shells (125mm) fired from the north-west. In both cases, the SMM was able to conclude the direction of fire to have been from the area of government-controlled Pisky (11km north-west of Donetsk) and Pervomaiske (17km north-west of Donetsk)

Note that the OSCE has confirmed the hospitals location and the type of shells used. Here is the hospitals location on google maps, as can be seen it is deep within the city and more than 7 km away from the front lines (Government controlled territory is even further away, 11-17 km as the OSCE suggests).

hospitalLocation.jpg

Now the explosive shells in question should be similar to these. They have a thicker cross section than the armour-piercing penetrator types, thus their aerodynamic properties would likely be worse. Furthermore, this type of ammo has a muzzle velocity of about 850 m/sec (while the armour-piercing round had 1800 m/sec), thus their range will be much lower than the armour piercing round. So based on the previous analysis and the reduced range of fragmentation shells we can deduce that a 11km or 17km range for these kinds of shells is impossible, even the front lines are too far away for the hospital to have been shelled from there. Thus it is technically impossible for the Ukrainian government to have shelled the hospital with these types of shells, the only possible explanation is that it was shelled by the militants from within their own territory.

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