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# How to use ballistic missile calculator

This post will show how to use the ballistic missile calculator, what the parameters mean and what values are reasonable for them.

Starting point are obtaining or determining:

• Stage diameter(s)

• The total missile mass and it's payload

• Missile length

The more difficult second task, involves reasonable estimates of following parameters, since this data is rarely openly available:

• Specific impulse of the stage(s) propulsion system

• The mass of the missile without fuel, i.e the empty mass

## How to get the data

Dimensions of missile are often available with good accuracy, sometimes official, because they can be rather easily derived from photo analysis.

Most dimensions on sites like Wikipedia, astronautix.com or Norbert Brügges missile enthusiast website are sufficiently accurate for analysis, especially of well known missiles.

More detail dimensions, like each stage length can be more difficult to obtain.

Data on mass is harder to get than dimensions, but as companies and nations often try to showcase the performance of the missile, by stating the maximum payload, at least this important parameter is also often known.

Still, estimates on the total missile mass are often available, but with variable degree of accuracy for realistic reverse-engineering.

Together with this data, often together with the payload, the maximum range of the missile is stated in different sources.

However for disinformation purposes the range - payload relation is often incorrect. A maximum payload that results in reduced range may be stated with the maximum range. This however, in reality, may only be possible with a reduced payload.

Determining this relation, the kinematic performance, is the main purpose of the ballistic missile calculator.

## Iterative reverse engineering

Therefore the range - payload relation is our starting point to calculate the two most difficult parameters left for a working missile model.

The Specific impulse or Isp, is a value which includes the average flight performance of the:

• Propellant

• Efficiency of the propulsion (e.g combustion chamber pressure, used power cycle)

• Nozzle efficiency

To get an understanding here are a few examples on Isp values:

Compared to the less advanced, ~235s Isp, Prithvi II engine, the R-17 Scud-B only achieves an Isp of just around ~230s. This is due to the jet vane thrust vector control system it uses and the thrust losses this technique causes. However, the same TVC system allows the R-17 to leave dense atmospheric layers faster and thus improves on its aerodynamic losses.

The 1950's technology Minuteman III first stage achieved a Isp of ~237s, while its 60's technology second stage reaches ~288s. This huge difference is due to the comparably inefficient 4 hinged nozzle TVC design of the first stage, the sea level launch/nozzle expansion ratio, and the less energetic solid propellant.

So Isp can be determined by the:

• Nozzle expansion ratio, whether it is first stage sea level optimized or for vacuum upper stages.

• TVC system used and number of nozzles

• Energy level of the propellant used, the most uncertain one of the three points

## The most difficult parameter

Determining the inert/empty/un-fueled stage mass is the parameter most difficult to obtain, for a realistic performance analysis. Obsolete exceptions, like the U.S Titan-II, excluded this parameter is next to never released officially.

The only way to determine it, is good knowledge on the technology used for the missile airframe, its layout and many other details.

So without this vast, highly technical detail knowledge, a different path is to obtain the empty mass is by iterative comparing it with the officially stated range and payload data.

Well known, mostly older missiles such as the R-17 Scud-B allow for such an approach. Once the empty mass is approximated, it can be reviewed whether it is reasonable, or how it compares to other missiles. This is a empirical approach to approximate a unknown parameter.

A resulting parameter is created, once the empty mass is approximated, the structural ratio (or deadweight fraction or dry-to-wet mass ratio).

In combination with the specific impulse Isp, these two parameters determine the kinematic performance level of the missile, i.e what we want to know.

The lower the structural ratio and the higher the Isp, the longer the range and the higher the payload.

The better these two parameters are, the more compact, light and thus survivable the missile.

## Things to check

Following points must be checked and confirmed for a missile model:

Thrust of the propulsion system, must be larger than the mass of the missile, commonly at least 20% higher.

The Isp, a vacuum optimized stage (nozzle) can achieve, will not be reached inside the atmosphere with the first stage. Hence this value must be increased or decreased according to operating environment/pressure.

Missile with a too high length to diameter ratio in a single stage, can get problems with the center of gravity and center of pressure relation. Hence very high l/d ratio stages are often technically not feasible.

Unusual fin designs, such as the grid fins on the Soviet Oka, or the large mid-wings of the Indian Prithvi should be replaced by a suitable "large rear stabilizer" when the visual missile builder module is used. The aerodynamic effect, the drag is whats important for calculation.

## The calculator

Best practice is to use the data from the missile database, of a comparable missile.

As next step, this data can be modified to reach the desired model.

• A parameter in the calculator that impacts the results is the aerodynamic layout. The higher amount and larger size fins and wings a missile has, the worse its aerodynamic performance

• The more blunt the nosetip and shorter/thicker the missile (l/d ratio), the worse its aerodynamic performance

Most missiles in the short range class have fins for stabilization and a tipped nose. Hence they can be approximated in most cases with the medium aerodynamic layout option.

The size parameters of the calculator have no important impact for the performance and are primarily there to allow a size estimate of the modeled missile.

Here the void factor is a correction parameter, to cover all uncertainties which can take "waste" a variable amount of space inside the missile.

Such spaces are primarily:

• Intertank section, the space between two liquid propellant tanks. Or Interstage sections, the space between two stages

• Tank or motor-casing bulkhead design and shape

• Nozzles integrated into the solid propellant motor casing to some extend. Or engines submerged in fuel tanks

• Propellant plumbing/pipes inside the tanks. Or the empty grain design used in the solid propellant motor-casing

and more.

Except for a visible guidance section, these difficult to quantify parameters are all collected in the void factor. So the void factor should be adjusted to approximate the known length of the missile, or taken from a similar, comparable missile.

The smaller the void factor, the higher the miniaturization and generally the overall quality of missile design and sophistication.

## Results

Of course, the key information a reverse-engineered missile model provides is the maximum range it reaches and the payload weight it is able to deliver.

However the results must be considered with an error tolerance to compensate uncertainties.

These can include:

• Trajectories other than the minimum energy trajectory, such as depressed ones, or maneuvers like terminal vertical attack

• Aeroballistic effect like a terminal glide phase for extended range due to additional lift

• Launch location (latitude and altitude) and launch direction (benefiting from earth rotation, or not)

• Uncertainties of aerodynamic and gravitational losses (acceleration, burn-time, time for staging process)

Due to these influences and more, an error margin of + - 10% of the total range must be assumed.

## FAQ

Q: Why are some of the data of the missiles from the database and calculator different?

A: The database values have been obtained by a more complex version of the calculator not released.

Q: Why are the shapes and wing/fin positions of some of the visual model of the data-base different to the real missiles? Is this due to a faulty model?

A: No, the automatic shape generating program is just not as advanced to display all possible missile shapes, especially rare and exotic shapes.

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