How does one know where a rocket is about to land? What affects its flight trajectory and how can a missile be aimed at it? The scientific challenges behind the Iron Dome

Recently we have been hearing again about rocket fire from Gaza, along with successful rocket interceptions by Israel’s Iron Dome air-defence system. How can a rocket be intercepted? How does one know where a fired rocket is about to land? What challenges have to be met in order to launch a missile to destroy it and how is this done nonetheless?

A rocket interception system is equipped with an excellent radar. The radar emits electromagnetic radiation in a certain direction. When this radiation hits a rocket, or another moving object in its path, it is reflected to a detector. Calculation of the time differences between the launch of the beam and its absorption by a detector enables identification of the whereabouts of the object, and in the case of a moving object, such as a rocket, a short follow-up also enables calculation of its trajectory. At that moment the system has to perform two tasks: It needs to quickly identify where the rocket is about to land and to calculate where a missile should be launched to intercept it along its trajectory.

Our first task will be to understand the expected motion of a rocket using physical principles. For the sake of simplicity, we will begin with the most explanation of basic rocket motion. We will then consider the real conditions in which rockets are launched and understand why the complexity of the actual trajectory renders its prediction and analysis challenging.

## Rockets and Trajectories

A rocket is a cylindrical motorized object, launched from a launch pad or from a dedicated launcher. Unlike a missile, a rocket has no guidance system, implying that it cannot correct its course during flight but rather follows a natural ballistic trajectory. Its upper part generally contains explosives, designed to explode upon impact and cause damage. This part is called a ‘warhead’. Many different types of rockets exist, differing in size, in the size and weight of the warhead as well as in their propulsion capabilities and hence their reach ranges differ as well.

To understand the trajectory of a rocket let’s assume that the rocket is a simple body. Its angle from the ground is determined upon lift-off and its engine pushes it at a constant force, from the moment it is switched on until it shuts down. The rocket then proceeds to move in freefall under the influence of gravity.

## Phase One: Take-off and Acceleration

First, a rocket is launched from a launcher aimed at a certain angle from the ground. It is propelled upwards by the engine at the launch angle and, unlike in the case of a shell fired from a cannon, the engine continues to operate for some time after the rocket leaves the launcher, causing the rocket to accelerate. Assuming the simplified model, we will ignore at this point the effect of air upon the rocket and focus solely on two forces that act upon the rocket - the force applied by the engine to propel the rocket upwards, and the force of Earth’s gravity pulling the rocket to the ground.

The acceleration phase. The rocket engine propels it upwards at the launch angle. Right: The forces acting upon a rocket.

## Phase Two: Freefall

Once its engine runs out of fuel and the rocket no longer emits a trail of fire and gases, its motion is governed by two forces only: gravity and friction with the air (termed ‘air resistance’ or ‘drag’), which we will ignore for now. When the engine stops, the rocket continues its motion at the angle and the speed that it has acquired up until the end of the acceleration phase. To understand a rocket’s motion we will project it onto a set of two perpendicular coordinate axes, such that motion of the rocket along each axis will be independent of its motion along its counterpart. We will term the vertical axis the Z-axis, and the horizontal axis the X-axis.

Movement along the horizontal axis is very simple. No force acts upon the rocket in this direction and it will therefore keep moving at a constant speed (V), similar to its speed at the time of engine shutdown, until it hits the ground, due to its motion along the vertical axis. The distance that the rocket will travel along the X-axis equals V times t (Vt, the time-of-flight). This time will be determined by the movement along the vertical axis.

Movement along the vertical axis is also relatively simple. The major force at work is gravity, causing the rocket to accelerate towards the ground at the gravitational acceleration (g), which is approximately 9.8 meters per second squared. Since the rocket was moving with upward speed at the time of engine shutdown, it will continue moving upward while losing speed, until it has reached its maximum height and will then start falling until it hits the ground. The rocket’s trajectory at this point is shaped like a parabola.

Second phase: Once its engine runs out of fuel, the rocket goes into freefall, affected only by gravity.

## Deviations from the Ideal Trajectory

So far we have described a fairly simplified trajectory. Seemingly, any radar that could identify the rocket during flight could also immediately calculate its trajectory to the point of impact. In practice however, a multitude of other factors affect rocket flight, complicating the situation considerably.

One of the most significant of these factors is drag, or air resistance. Two main aerodynamic forces are involved in the motion of a rocket through the air. The lift force, which is very significant for large and winged aircrafts such as airplanes, but negligible for a small rocket, and the drag force, generated by the friction of the rocket with the air, which, in its turn, depends on the velocity of the rocket and is therefore not constant. The faster the rocket moves, the greater the friction that acts upon it, acting always in the direction opposite to the direction of movement.

Drag can be reduced by adequate aerodynamic rocket design, but it cannot be entirely eliminated. This force affects the relatively simple motion we described earlier: it slows the rocket down, making it reach a lower altitude and fall closer to the point of lift-off. Drag also depends on air density, which is not uniform along the rocket’s trajectory and varies according to the altitude.

## Torque

Thus far, we have treated the rocket as if it moved on a two-dimensional plane - described by vertical (Z) and horizontal (X) coordinates. A third axis (Y) also exists, but our analysis so far did not not include a force that could act upon the rocket in this direction. We have treated the rocket as a simple structureless body, such as a point or a line. In reality, however, a rocket has a complex volume and structure. Therefore, the forces exerted upon it during flight are not uniform, such that one side of a rocket might be affected stronger than its counterparts. This will lead to a ‘final moment of force’, or ‘torque’, which will strive to rotate the rocket and divert it off its course.

To understand why non-uniformly distributed forces can drive a rocket off its course, imagine the following situation: Suppose you are standing with your arms spread to the sides and two friends are pushing you backwards by the palms of your hands. If they push with equal force, you will fall flat on your back, but if one exerts more force than the other, you could find yourself rotating as well as falling backwards. One side of your body is supposedly trying to move faster than the other. The exact same thing happens to a rocket when the forces exerted on it from all sides aren’t uniformly distributed.

Such deviations may stem from a non-ideal launching pad, an imperfect engine structure, or air resistance, in case that the friction is not uniform in the different parts of the rocket. Rocket structure, manufacturing defects and weather conditions (winds, depressions, etc.) are other major influencing factors.

This phenomenon is far from negligible, posing a significant challenge to rocket developers. One common way to deal with this is by adding fins at the sides of the rocket. The air flowing and rubbing against these fins causes the rocket to spin around itself as it flies. This self-rotation helps to minimize the deviation in the trajectory caused by an imbalance of forces, by ‘dispersing’ the imbalance evenly in all directions.

Even after the addition of fins, rockets can never be completely accurate and there will always be some uncertainty concerning their trajectory. Even sophisticated rockets, such as the Iranian Fajr, suffer from inaccuracy. A Fajr 3 rocket, launched at a target 40 km away, will hit the ground within a radius of up to one kilometer from the target it was aimed at. The exact impact site will be greatly affected by the conditions encountered by the rocket during its flight.

Successful rocket interception by the Iron Dome during Operation Protective Edge (2014, video by the IDF):

## Detection of a rocket, calculation of its trajectory and interception

Based on this knowledge, we can begin to understand the challenges facing a rocket interception system such as the Iron Dome.

The first step is to track a rocket's trajectory. At this stage, the radar of the system needs to detect the rocket and rapidly compute its speed and trajectory. A rocket is likely to be detected immediately after being launched, but since trajectory calculation requires knowledge of the rocket’s velocity at the beginning of freefall it is next to impossible to estimate where the rocket will hit as long as its engine is running. If the radar is able to detect the rocket type and is familiar with its characteristics, the system may be able to predict, already at this stage, the time required for engine shutdown. Without this prior knowledge, it is not possible to predict the rocket trajectory as long as it has not completed its acceleration.

As soon as freefall begins, the trajectory of a rocket is easy to compute using the information of its altitude and velocity, which is constantly provided by the radar. In practice however, such a calculation is not sufficient since it does not take air resistance into account. Therefore, a different way must be found to accurately predict the trajectory of a rocket.

At this point, sophisticated algorithms kick in, which certainly constitutes one of the most important components of the Iron Dome system. One way to improve the accuracy of trajectory prediction is to estimate the effect of drag based on the deviations of the rocket from the ideal trajectory calculated at the beginning of its motion. The system can thus quickly identify the rate at which the rocket loses momentum due to its friction with the air (i.e. the decrease in velocity and altitude), and the trajectory can thus be re-evaluated.

In the operational system, sophisticated algorithms must constantly recalculate the rocket’s trajectory, based on the data supplied by the radar. It is most likely that they not only identify the effect of drag on the trajectory but also incorporate historical data of known rocket trajectories into their calculations.

After a rocket’s trajectory is computed, it is important to understand whether the rocket is expected to hit an open area or whether it should be intercepted. Weighting all the available information and taking into account the parameters on the intercepting missile, the system can find the optimal place to intercept the rocket.

It is important to note that even the best radars are not entirely precise in their analysis of a rocket’s trajectory. It is therefore impossible to pre-plan the precise trajectories of the intercepting missiles so as to hit the approaching rocket directly. Hence, and in accordance with the information that has been published about the Iron Dome, the intercepting missiles are launched into the rocket’s estimated trajectory but constantly receive corrections from the radar and are also equipped with their own tracking devices that enable rocket identification and trajectory adjustments.

Even following the above mentioned corrections, intercepting missiles do not hit a rocket directly, but rather get as close as possible to it and explode nearby. The blast wave and shrapnel of the intercepting missiles are expected to hit the warhead of the enemy rocket, causing it to explode while up in the air.

## Summary

We have seen that the motion of a rocket depends on relatively simple forces, such as gravity and drag, making its trajectory relatively easy to compute and predict using simple physical principles. However, in the absence of prior information on the aerodynamic structure of a rocket and specific details of the drag and torque forces that act upon it, the task at hand becomes significantly more complex. Thus, optimal analysis of rocket motion requires continuous tracking of the rocket and constant refinement of its trajectory calculation and prediction, using sophisticated algorithms.