I've been pondering about the range thing, because my immediate kneejerk was "two balls rolling, one's 2x faster than the other, that one will go further" … but I had that "trick question" flag go off as soon as I thought that.
Things I don't know:
- It's faster … possibly huge fuel consumption to get to operating speed over comparable subsonic missile? (even if sips fuel once at operating speed/altitude)
- Will fly at higher altitude … which means lower friction (enough to counter what's added by higher speed) … but also to hit the same target, more distance must be covered (climb/descent to operating altitude + greater orbiting circumference)
- Limited "package" size … do scramjet surfaces/elements reduce the amount of space available for fuel?
Here's how one can think about this, using numbers.
First, specific impulse [1]. It's measured in seconds, and it means how many seconds an engine can provide 1 pound of thrust if it burns 1 pound of fuel. The graph [2] shows that specific impulse is a decreasing function of speed, with a profile that looks roughly like a hyperbola. The specific impulse of the engines powering Boeing 747 is listed at 6000 seconds. The specific impulse of a scramjet engine at Mach 7 is plotted at about 1000 seconds. Finally, the specific impulse of the Space Shuttle solid rocket boosters is 250 seconds (it's down in the Examples subsection).
The specific impulse of a subsonic cruise missile is not listed. However, the engine of both the AGM-86 and the Tomahawk cruise missile, the Williams F-107 [3] has a bypass ratio of 1:1, which is much much lower than the bypass ratio of commercial jets [4]. I'll make up the number to be 2500 seconds (it will make my computations below a bit easier, you can adjust them accordingly, but you'll see that it does not make a huge difference if it's 2000 or 3000 seconds; I don't think there's a chance it's above 3000 seconds).
Second thing is the lift-to-drag ratio. For hypersonic vehicles, there's the empirical observation that the max L/D ratio is 4(M+3)/M, where M is the Mach number. If M=7, we get 40/7 = 5.71. For subsonic vehicles, wikipedia lists all sorts of L/D ratios, but they roughly range from 10 to 20.
The last is the rocket equation [6] which says that delta-v is the specific impulse times the gravitational acceleration times the log of the ratio of the initial mass (i.e the fully fueled rocket) and the final mass (the "dry" rocket, after the fuel was burned).
Let's compare now a subsonic and a hypersonic missile. The subsonic one travels at Mach 0.73 (like the AGM-86) and the hypersonic 10 times faster, at Mach 7.3 (I'm making this up of course, nobody has any idea how fast the Raytheon missile will be, except that by the definition of "hypersonic" it needs to be above Mach 5). In meters per second, those speeds are around 250 m/s and 2500 m/s.
Let's also say they both have a "dry mass" of 900 kg (the mass of the warhead and the engine and the vehicle itself, but excluding any fuel).
And let's say the missile has to have a range of 2500 km.
The L/D ratio of the subsonic missile, we'll take it to be 15 (most likely it's lower than that, but I don't think it would be less than 10). The L/D ratio of the hypersonic one should be lower than 5.71, we'll take it to be 5.
The hypersonic missile will need 1000 seconds (about 17 minutes) to travel the 2500 km distance. Since it has an L/D ratio of 5 and a mass of 900, it will have a drag of 900/5 = 180 kg. The specific impulse of the hypersonic missile is 1000 seconds, so you need exactly 180 kg of fuel to push agains the 180 kg of drag for the 1000 seconds of the trip. Of course, you need some fuel to push that fuel, but this is second order things, and we can ignore.
The subsonic missile will travel for 10000 seconds (about 3 hours). It will have a drag of only 60 kg, and it has a specific impulse of 2500 seconds. You end up with a fuel need of 240 kg. That's more than for the hypersonic, but then we are not done with the hypersonic, since that one needs a rocket booster to get to its cruising altitude and speed.
Going to an altitude of 100 km is equivalent to gaining a delta-v of 1400 m/s (this comes from the kinetic energy =mv^2/2 and potential energy = mgh). We need another delta-v of Mach 7.3 = 2500 m/s, that's a total of 3900 m/s. The specific impulse of the booster is 250, and we take the gravitational acceleration to be 10, so the log of the mass ratio is 3900/2500 = 1.56, which means the initial mass is exp(1.56) = 4.76 higher than the dry mass. To push 1000 kg of hypersonic missile, the booster needs to have 3760 kg of fuel. We can ignore the dry mass of the booster itself. In any case, we need a ratio of roughly 4:1 of booster/hypersonic vehicle.
All in all, it looks like a hypersonic missile is, if anything, better in cruise mode than a subsonic missile, but since it needs the huge booster to get to cruise mode, it looks like it will carry about one fifth of the payload for the same range.
Which is not so bad. If you carry 100 kg of explosive instead of 500 kg, you can still do a lot of damage.
Please tell me you had some experience with this stuff before busting out a dissertation on of lift-to-drag calculations …
So, if we're constraining ourselves to go the same distance as a cruise missile with the same max weight/dimension, we CAN, but …
- using less fuel
- and at 10x speed
- but have a smaller warhead
So to your original question about whether a scramjet will travel further on the same amount of fuel, the answer does appear to be YES. (But with a smaller warhead, assuming same max weight/dimensions of conventional cruise missile)
So I guess the next question is: if we assume distance and dimension are constraints and we _do_ have an 80% smaller warhead … do we get 80% less damage when it impacts, given that it's traveling at 10x? Or does the speed impact the force of the warhead?
> Or does the speed impact the force of the warhead?
Absolutely. The energy density of TNT is 4.11 MJ/kg. An object traveling at 2.5 km/s has a kinetic energy (mv^2/2) of 3.12 MJ/kg. If we consider a simple rule of 3 for a cruise missile (1/3 the warhead, 1/3 the fuel, 1/3 the rest of the missile), then the kinetic energy of the warhead and the empty vehicle will be 50% more than the energy of the explosive (if not more). So, even if the warhead is only one fifth of the warhead of a subsonic missile, the total energy delivered on target is one half.
