missile speeds and ballistic missile question

1. We know that the average BVR AAM is usually qouted with max speeds of some mach 4, perhaps going up to mach 5. Doesnt matter if its amraam, sparrow, mica or r-27. Now those speeds are usually qouted as theoretical maximum, when used at high altitude. Much thinner air means the given impulse/drag ratio of the rocket motor will give the missile greater speed which will give it greater energy, thus giving it greater range.

It is also said the same missile fired near sea level will have much shorter range. The difference in air density at sea level and at, say, 40.000 feet, is some 250%, does that mean that the maximum speed of the missile is also going to be reduced by that amount? Sea level density is 1, 40.000 feet density is little under 0.4 How much will max speed be at sea level if it was mach 5 at 40.000 feet? mach 2? Or something else?

Ok, let’s talk about projetile (missile) that achieves M4 at 40k ft.
So, at M4 (40k ft), such a projectile will have skin temperature of ~650°C and a Dynamic Pressure (essentially drag) it needs to overcome of ~210KPa. Since the temperature in tropopause is constant, we’ll work with Dynamic Pressure.
Now, if the launching aircraft descends to 0 ft (sea-level) and fires the same missile, it will most definitely have reduced range. The projectile speed that produces Dynamic Pressure on 40k ft of 210KPa would be ~M1.7, at sea level.

So yes, the M4 at 40k ft projectile would have top speed of ~M1.7 at sea level (outright speed depends on launching aircraft) and average mission speed much lower. The impact in range is even greater and falls to 1/3 against optimal (non-maneuvering) target, or way less, essentially putting AMRAAM (or other MRAAM) in WVR class weapon, at most extreme conditions.

This calculation has been given for standard atmosphere and generic missile and particular models will have different outright performance.

I believe the max speed of the missile depend more on the speed of the launching a/c then the altitude, but the missile will drop in speed even much more rapidly at low altitude.
High altitude on the other hand gives the missile fins very poor authority with a burned out engine.

Once, outside the atmosphere, you can safely predict the speed based on Keplerian movements laws. That’s, if you neglect the Earth curvature, on simple ballistic laws. Drag during boost phase applies with considerable effect just to SRBM or MRBM with depressed trajectories.

I believe the max speed of the missile depend more on the speed of the launching a/c then the altitude, but the missile will drop in speed even much more rapidly at low altitude.
High altitude on the other hand gives the missile fins very poor authority with a burned out engine.

Would i be then correct in assuming that a similar, albeit smaller ballistic missile with 500 kg payload, having a range of 2000 km, would hit apogee around 450 km altitude, its speed would reach 4 km/sec in around 60 seconds, then drop to 3 km/sec around 300th second, accelerate once again to 4 km/sec around 550th second and hit its target around 570th second with impact speed of around 1,2 km/sec?

I can’t give you precise figures for a 2,000 km range missile with a 500 kg payload, but modelling of Iran’s Sejjil solid-propellant ballistic missile with an assumed 1,000 kg payload gives the following figures:

Second stage burnout at 100 seconds into the flight – velocity 4.25 km/sec.

Apogee at around 450 sec, at an altitude of 535 km, a ground range of 1,200 km, and a velocity of 3.2 km/sec.

Speed building up during the descent, reaching a peak of about 4.4 km/sec at an altitude of 32 km.

Impact velocity of 2.7 km/sec at 830 sec.

Here something from a paper on Indian BM

http://img233.imageshack.us/img233/9119/typicalpaths.jpg

These charts are a useful summary of the basics of ballistic-missile trajectories. I could use the first of these next time I have to teach the basics of BM flight. It neatly summarises the information expressed in a lot of range v velocity tables in my files. I’ve tracked down a copy of the original paper, so will be able to cite source details. However, the only copy on-line now seems to be an HTML version (less graphics) created by Google.

I guess that’s it, then. I’ve got no more questions concerning ballistic missiles. Thanks, Rodolfo. There’s only the first question left, unrelated to ballistic missiles.

1. We know that the average BVR AAM is usually qouted with max speeds of some mach 4, perhaps going up to mach 5. Doesnt matter if its amraam, sparrow, mica or r-27. Now those speeds are usually qouted as theoretical maximum, when used at high altitude. Much thinner air means the given impulse/drag ratio of the rocket motor will give the missile greater speed which will give it greater energy, thus giving it greater range.

It is also said the same missile fired near sea level will have much shorter range. The difference in air density at sea level and at, say, 40.000 feet, is some 250%, does that mean that the maximum speed of the missile is also going to be reduced by that amount? Sea level density is 1, 40.000 feet density is little under 0.4 How much will max speed be at sea level if it was mach 5 at 40.000 feet? mach 2? Or something else?

Roughly speaking, …yes.

Anyway, for a 2000 km launch you can neglect the Earth curvature. So, you should use the ballistic laws to simulate the trajectory in order to obtain more accurate data. I.e, you can assume a burnout time around 120 secs, a launch angle of 45°, constant acceleration and a final speed of 4 km/s. The rest of the path will be a parabola very close to the real path, until, of course the re-entry stage, around 100 km height. Varying the launch angle you can simulate depressed or lofted trajectories.

Once again, thank you. Would i be then correct in assuming that a similar, albeit smaller ballistic missile with 500 kg payload, having a range of 2000 km, would hit apogee around 450 km altitude, its speed would reach 4 km/sec in around 60 seconds, then drop to 3 km/sec around 300th second, accelerate once again to 4 km/sec around 550th second and hit its target around 570th second with impact speed of around 1,2 km/sec?

Here we go:

http://img171.imageshack.us/img171/8254/speedrange.jpg

http://img11.imageshack.us/img11/7705/4000kmpath.jpg

Scaled down? Yes.
Scaled up? No; because of the Earth curvature. That’s cause the path to be like an ellipse rather than a parabola.

Let me some time, I will up-load a better resolution graphs of the same figures.

Thank you! That graph is great, i was hoping for something like that, which would give me a whole range of values. I noticed the max velocity (for the given range) in the first graph is the same as the velocity in the second graph (for the same range). Do you think one could linearly scale down and scale up other range values?

Here something from a paper on Indian BM

http://img233.imageshack.us/img233/9119/typicalpaths.jpg

Thank you, Rodolfo. I shall certainly try to find more precise data and, if i manage to do that, I will share it here.

I think I can provide you a qualitative but a little inaccurate answer. By the sake of simplicity we can divide the ballistic path in (1) an exo-atmospheric phase. (2) An endo-atmospheric phase in the upper layer of the atmosphere (100-40 km altitude) and (3) an endo-atmospheric phase in the thick lower layer of the atmosphere (less than 40 km). In (1) you can accurately compute the vehicle position, speed and azimuth from Keplerian motion laws. In (2) these laws begin to be interfered by drag forces and in (3) drag “rules”. Speed in steps (2) and (3) ends to depend on motion parameters and depends mainly on the ballistic coefficient of warheads (that are unsurprisingly classified information). So let us assume that we have a “heavy” RV and a “light” RV, both descending from 100 km to the ground. The heavy one can descend at around 4 km/s (from a ballistic speed of 7 km/s) for most of the last 100 km while the light one at an average speed around 2 km/s. Below, let say 10 km, both travel around 2 km/s but on the average the heavy one descended much faster. Numbers are guesses.
I don’t know the values for ballistic coefficient of sub-munitions but we can safely assume they are “light”. Sub-munitions being small light RV will have low descending speed in steps (2) and (3). We can’t predict much without data on ballistic coefficients. In addition depressed trajectories increase the travel time on steps (2)-(3), so more drag, and so less final speed.

Off- course this is a very simplified picture of the reality.

The initial speed of the missile depend on the speed of the launching aircraft, while altitude and maneuvering will determine energy bleed, where some missiles/control surfaces are draggier then others.

Thank you for your time, Rodolfo. Sadly, I didnt see any concrete speed info in that paper, and the references are unatainable for me, since theyre all paper publications.

Hopefully someone will join in and share their knowledge…

Here you have a typical paper comparing filters performances. References may also help you.

http://www.eurasip.org/Proceedings/Eusipco/Eusipco2006/papers/1568981953.pdf

Interesting questions. There are some equations that can predict the drag for a reentry-vehicle as function on the altitude, the vehicle speed and its ballistic coefficient. The problem is “a classical one” on the nonlinear tracking literature. Let me some time to google for internet sources on the issue.