VictorInThePacific
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Final version of the intercept calculations.
The YJ-83 missiles can be launched from outside of SM-2 range. There is no advantage to launching them from closer in (but see below). However, there may be an advantage to launching the YJ-91 ARMs from closer in than their maximum range. Because these missiles travel at low altitude or higher, the ships can see them further out than the sea-skimmers, and can fire at them more often.
There are 5 ways of getting the missiles in closer.
1) Just fly a whole bunch of planes in to short range at any altitude and ignore the casualties. With a billion or so people to call on, the Chinese can probably afford to throw away their pilots and maybe even their toys, and their military history over the past 100 years (or always) shows that this is exactly the approach they do use. I do not recommend this approach.
2) Fly the planes in at any altitude and launch missiles just before the planes are shot down. This is better than the first option, because it does have the effect of absorbing more SAMs than just the missiles alone would do. I do not recommend this approach.
3) Fly the planes in as close as safely possible at low altitude and launch from there. Warhorse has given us the radar horizon from low altitude to a medium ship as 51 nm. For my convenience, I will take the safe low altitude to be 50 nm (note 1). If a SM-2 shoots at such a plane, the plane will always be able to escape. This range does not give a significant advantage for the YJ-91 ARM (in fact, that is probably its practical launch range), and it gives no advantage at all for the YJ-83. It would give a definite advantage to long-range cruise missiles.
4) Fly the planes in as close as safely possible at VL altitude and launch from there. Warhorse has given us the radar horizon from VL altitude to a large ship as 26 nm. For my convenience, I will take the safe VL altitude to be 25 nm (note 1). If a SM-2 shoots at such a plane, the plane will always be able to escape. Most planes cannot safely fly at this altitude. For such planes, I do not recommend this approach, because even though it may save on missiles, I consider the cost in pilots, even if slight, to be far higher than any number of inanimate objects. (Ed: What about human lives on the other side?) However, Warhorse has pointed out that the Su-30MK2 DOES safely fly at VL. This is deadly, deadly stuff. These planes could just park at VL 25 nm from the ship; ordnance hurled from that range would give the ship very little time to respond, although how exactly such a plane would loiter at wavetop height is a bit of a mystery.
5) Fly in on an oblique approach. I have discussed this earlier and elsewhere. For example, see my post of 2/11/09 in my thread "outmaneuvering SAMs". I am not going to discuss this further here, because the present subject is already complex enough.
According to Brad's post 47, this thread ("I think the SM-2's could be launched on inertial/midcourse guidance before they enter the ship's own radar LOS, assuming that another asset (such as the E-2C Hawkeye with CEC) could "see" the targets."), the ship does not need to have radar LOS to the target to launch the SM-2s if a friendly unit can see the target, although the ship still needs to have the target on its own radar at the time of intercept. This means that the Chinese planes could safely park at 25 nm (VL) or 50 nm (low). On the other hand, for inbound missiles, which are definitely going to enter the ship's radar LOS, an intercept course would be calculated that would let the intercept happen at those ranges. (Exactly? I doubt it. How far in? I have no idea, so I am using the stated figures.) However, bear in mind that a missile does not actually exist as a target until after it has been launched, which means that a missile carried in by a plane has a definite advantage over one coming in on its own steam.
So then I have to raise the following issue the second time. Based on the data Warhorse has provided, Chinese fighters with 40-odd nm range AAMs are parked at VL 25 nm from the nearest US ship, and 30 nm from the center of the formation. Where can the Americans have air assets with radar to look down on the battle area? Nowhere, I think. Nevertheless, we are assuming that those assets exist.
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Finally, we get to the calculations.
a) YJ-83 cruise missile
This missile comes in at VL at 990 kn. I am using 1,000 kn in the calculation, which is only 1% different, makes my life a whole lot easier, and gives results that are as good as the exact calculation. (note 1) Since this missile is coming in from far away, it has been detected some time ago and an intercept has been calculated to take place at 25 nm. 1,000 kn means it travels 1 nm in 3.6 s and 25 nm in 90 s. The SM-2 moves twice as fast (2,000 kn) and travels 25 nm in 45 s. This means that the SAM launcher has cycled by the time the first intercept happens.
Intercept 2: The missiles are moving with a combined speed of 1,000 + 2,000 = 3,000 kn. It takes them 30 s to reach the intercept point, which is 25 nm x 2/3 = 16.7 nm out.
Intercept 3: There is no launcher delay. With only 2/3 of the distance to cover relative to the last step, the time to intercept is now 20 s. The intercept happens 25 nm x 2/3 x 2/3 = 100/9 = 11 nm out.
Intercept 4: There is a 10 s launcher delay. There are 30 s left on the clock, so the YJ-83s are 25 nm x 1/3 out. Time to intercept time is 10 s, and it happens 25 nm x 1/3 x 2/3 = 5.5 nm out. There is no minimum range problem.
No more intercepts are possible, because the SAM launcher will not cycle in time.
This is better for the US than what I calculated on 2/20 (post 39), because I am now utilizing the AWACS.
b) YJ-91 ARM launched from 50 nm, low
This missile comes in at 1,940 kn. I am using 2,000 kn in the calculation, which is only 3% different, makes my life a whole lot easier, and gives results that are as good as the exact calculation. (note 1) Since this missile has just been launched, the SAMs are still in their launcher. I am giving the ship an automatic detection, because there's a whole batch of incoming missiles, and the ship should be able to pick up some of them. 2,000 kn means the YJ-91 travels 1 nm in 1.8 s and 50 nm in 90 s. The SM-2 moves at the same speed (2,000 kn). Therefore the first intercept happens at 45 s and 25 nm out.
Intercept 2: There is no launcher delay. The intercept happens 22.5 s later and 12.5 nm out.
Intercept 3: There is a 7.5 s launcher delay. There are 15 s left on the clock, so the YJ-91s are 50 nm x 1/6 = 8.3 nm out. The intercept happens 7.5 s later and 4.1 nm out. There is no minimum range problem.
No more intercepts are possible, because the SAM launcher will not cycle in time.
The main reason the ship only gets 3 intercepts in this case is that the missile did not exist as a separate entity until it was 50 nm out.
c) YJ-91 ARM launched from 25 nm, VL
I initially thought that this was impossible, for the following reason. The YJ-91 has a minimum flight altitude of low, and I assumed that there was a reason for this, such as if the missile gets too close to the water, it crashes. So while the Flanker could safely be there, the missile couldn't.
I have since read something that confirms this. However, there is nothing to prevent the Flanker parking 25 nm out at VL, and then, when the time is ripe, popping up to low altitude to shoot, and dropping back below the horizon. This is deadly, deadly stuff.
The missile comes in at 1,940 kn. I am using 2,000 kn in the calculation, which is only 3% different, makes my life a whole lot easier, and gives results that are as good as the exact calculation. (note 1) Since this missile has just been launched, the SAMs are still in their launcher. I am giving the ship an automatic detection, because there's a whole batch of incoming missiles, and the ship should be able to pick up some of them. 2,000 kn means the YJ-91 travels 1 nm in 1.8 s and 25 nm in 45 s. The SM-2 moves at the same speed (2,000 kn). Therefore the first intercept happens at 22.5 s and 12.5 nm out.
Intercept 2: There is a 7.5 s launcher delay. There are 15 s left on the clock, so the YJ-91s are 25 nm x 1/3 = 8.3 nm out. The intercept happens 7.5 s later and 4.1 nm out. There is no minimum range problem.
No more intercepts are possible, because the SAM launcher will not cycle in time.
Only 2 intercepts! The ship can only fire 40 SAMs and destroy 20 ARMs. After that, the ARMs start hitting and disassembling the ship. It doesn't matter which US ship is under attack; none of them get to expend their full complement of SAMs.
d) combination attacks
If the ship radars are off, it is safe against ARMs. I don't know if the ship can protect itself against ARMs already in flight by turning its radars off. If it can, it will be necessary to keep enough of the bigger missiles coming in so that the ship can't reasonably turn its radars off. In that case, the ship will be able to use some of its SAMs against the slower missiles in addition to two batches against the ARMs.
e) Enhanced Sea Sparrows
Although the ESSMs are faster and have a smaller minimum range than the SM-2s, they do not provide a useful advantage (more intercepts) in this situation. Anyway, there aren't very many of them, and they have a rather short maximum range.
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Notes:
1) On the merits of calculations with approximate figures.
The first thing we need to realize is that, no matter how many significant figures our calculator gives us, no number in real life can be considered exact, as long as it relates to a measurement. Each number has some uncertainty associated with it, as well as a certain number of significant digits, and those 2 concepts are linked.
There are other reasons why stating an exact number may not be meaningful. For example, in Harpoon, the radar horizon from a medium ship to a low altitude plane is stated as the exact number 51 nm. But how high, exactly is the radar emitter/receiver? And, more importantly, exactly how high is the plane? The 51 nm could only be exactly correct for one specific plane height for each specific radar height, and the value of 51 nm is therefore only a representation of the actual radar horizon.
Another example is how Harpoon handles detection. Detection attempts only happen at discrete 30 s intervals, and those are fixed with respect to the game clock. Now, I am not arguing against that system. But it needs to be realized that this procedure guarantees that many results can only be approximate.
Indeed, the whole game is based on calculations done by a computer algorithm at discrete time intervals. All numerical analysis is inherently approximate, even though the results can be and often are stated to umpteen significant figures.
There are 3 main reasons I am working with approximate numbers in most of this thread.
a) Accuracy. The important thing I need to work out in this calculation is how many intercepts happen for a given ASM/SAM combination. Number of intercepts is an integer, generally a small integer less than 5. But the intermediate results are intercept locations or times, which are real numbers. Suppose that the exact calculation tells me that the number of intercepts is a number anywhere between 3.2 and 3.8. This means that exactly 3 intercepts can happen. (A fractional intercept means that the launcher hasn't cycled yet, or that the intercept is too close to the ship to actually work.) Now if the approximate calculation tells me that the number of intercepts is anywhere in that range, then I have obtained EXACTLY the same USEFUL answer as with the exact calculation. 3.5 + .3 = 3.5 + 10%. Generally, if my approximate calculation is within 10% of the exact calculation, there is no meaningful loss of accuracy. In some special cases, the result is borderline, so I might need to recalculate more carefully.
b) Transferability. Using a representative number such as 1800 kn for my calculations, I obtain a useful result for a 1940 kn missile. But I can take all the results and immediately apply them to any missile moving at speeds from 1600 to 2000 kn. There is no need to recalculate anything.
c) Convenience. Especially in this particular post, you will see that I have done the entire calculation in my head, without once ever needing to use a calculator. This saves me a great deal of time. For example, if a missile travels at 1800 kn, which is 1800 nm per 3600 s, then it needs exactly 2 s to go one mile, and that is a very easy figure to work with. If it is travelling at 2000 kn, which is 2000 nm per 3600 s, then it needs exactly 1.8 s to go one mile, which is a fairly easy figure to work with. But if I have to use 1940 kn, every single calculation requires me to punch a bunch of numbers into my calculator. Incidentally, 1940 kn is within 3% of 2000 kn and 8% of 1800 kn, so both approximate figures will give excellent results.
Did I mention round-off error? Calculators can easily introduce round-off error if you're not careful.
The points I have raised here are standard for any numerical analysis. It's not a matter of me being lazy or sloppy. Of course, any one who wants to can use the calculation steps I have shown and recalculate with the exact values. I think that you will get the same results as the approximate calculation.
< Message edited by VictorInThePacific -- 2/25/2009 1:17:42 AM >
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RE: air attack on US SAG - 
The Kh-29 is a short ranged weapon (about 7.6 nm with the longest ranged variant), and Harpoon assigns a speed of 791 kt.
and this is what they say about that missile: 
Its already showing signs of that.
We are going to ignore all those details and assume that someone else has done those calculations, and now the airplanes have arrived at the target on schedule. I include all these details when I say "park" your airplanes at such-and-such a location. 
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