# How quickly could a spaceborne missile accelerate?

Is there any practical g-limit for unmanned spacecraft, or could you theoretically push a missile to significant fractions of c in hours or minutes while pulling hundreds or thousands of Gs? I need to design a relativistic kill vehicle that can be readily used for ship to ship combat without being intercepted (these are REALLY big ships).

A railgun is still on the table but missiles don't risk damaging the ship the same way a railgun discharge does, allowing for higher yields. Missiles need time to accelerate out in space independently of the ship however, which risks them being detected and destroyed before they can accelerate up to a speed at which counterattacking or even detection would be impossible.

So out in the vacuum of space, how many Gs of acceleration could you subject a large hunk of solid mass attached to antimatter-catalyzed fusion rockets with a targeting system and an explosive charge before you risk compromising the missile's structural integrity?

• Depends on the type of engine, the higher the efficiency to convert propellant into thrust(force) the better and no cap for ISP(specific impulse). Commented Oct 22, 2016 at 3:03

There isn't a meaningful limit in G's as you describe. The real G limit is based on the limits of the rocket fuel, but if you're hand waving that away with antimatter, then the only real limit is the ability of your materials to transmit force. If you design to that, you can make a missile which flexes into a strong stable shape under load.

The real challenge in these situations is vibration. If you're accelerating at 1g, having 5% of your energy going towards vibration is a minor annoyance. If you're accelerating at 100g, suddenly 5% of your energy is 5 G's. That's 5 G's in any arbitrary direction, not just the direction you made your rocket strong in.

So your real limit is going to be very dependent on the quality of your antimatter-catalyzed fusion rockets.

• Adding to the point on vibrations: look up pogo oscillation.
– user
Commented Oct 22, 2016 at 10:46

Acceleration depend on Thrust to Weight Ratio (TWR, examples)

• Merlin 1D rocket engine, full-thrust version - is capable 1765 $$m/s^2$$ , but it does not include mass of fuels and rest of construction.

Second limiting factor is ISP (and power generated) - which determines needed amount of fuel to achieve desired speed.

There are fuel less acceleration system, for example, solar sail like solutions and acceleration thing with laser - where just flat mirror-like surface may act as engine - it will have high ISP but also have some TWR which depends on perfection of the system.

Close to 1 c in one hour is pretty high acceleration about 80000 $$m/s^2$$ or about 8000 g - it is high, but not astonishing high even for ordinary materials (this depends on the form of construction). (As an example, 0.1 x 0.1 x 0.2 m piece of steel (2L) first liter of steel will press on second liter of steel with force about 56 tonne, whole acceleration force is 112 tonnes - this form and material is perfectly capable to withstand that force. Sure such bar of steel is not a rocket, and rocket with given acceleration rate will make hard times for developers to figure our how to build it).

# Tactics

Sad but it needs to be said, but you started on the wrong foot.

First of all you should think about tactics and what capabilities you have.

Missiles will be always capable for higher acceleration rates than ships, not because of humans, but because it may have no payload except engine, fuel, energy source, where ship should have all that and payload - humans, cargo, whatever. Missile be faster than ship is actually all you need in general. Antiship missiles are not the fastest of their kind, they are just enough fast for the task.

One hour to 1 c speed means actually an acceleration distance of about 540'000'000 km or 3.6 AU. - I personally find such combat distances pretty realistic, but what about the ships? Are they capable to break the distance? I guess not, but depends on the tech, and how prepares they are against attack and their goals. If not, nothing significant will change if you will hit the target not in one but two hours from four times of that distance. It may make no sense, but this point will be dealt with in next paragraph.

If you rely on speed of the rocket as means to protect it, and make harder for attacker to defend against it, there is a problem - it is not so simply. Thin foil around the ship, let say few light seconds around the ship - will effectively protect it from 1 c projectile, from any direction.

Big enough gas cloud (balloon or just a cloud either from the blast of antimissile's rocket exhaust or by exploding it (as example)) may act as effective countermeasure especially if you know the approximately direction and the time of launch of the attacking missile. (There are countermeasures for missile in that situation, but there are countermeasures for countermeasures.)

And for reasons of how thick should be that foil around the ship - thickness of it grows proportionally as the square of speed (proportionally to kinetic energy of projectile). For example, if foil with a 1 μm thickness is enough to destroy the projectile at 0.9 c, then it have to be 100 times thicker for a 0.09 c projectile and this means 100 times more mass for that shell and this may make this sort of defense ineffective. This is an example of how lower speed may affect defense strategies making it more complex for defending vessels. Yes, you may have difficulties to detect 1 c projectiles with your systems, but you do not have to detect them to be protected against them. Lower speeds of projectiles allows the use of detection systems but what it actually means is replacing a simple at-all-times working solution, with a more complex and less efficient solution (which may not work).

There is delicate balance between many of the factors, and depending on particular implementations there may be some local optimum for speed differences between the target vessel and missile, but it is not necessary have to be 1 c velocities, and most likely not for ship-to-ship combats at multiple AU distances.

How big the ships are is actually not very much important, as far as with their sizes - missiles and defense systems scale proportionally to the mass of the ships.

There is an example of an attempt to model possible tactics for combat in the solar system. But if you prefer combat in interstellar space it will work in a similar way just replace the star with a ship and adjust scales to match the sizes. (detection and countermeasures swarm shell may be still light hours around a ship)

So I recommend starting with combat tactics, with the ships' characteristics, with the nature of the space where the combat takes place, and with goals of defenders and attackers.

The nature of tactics is very sensitive to details, the slightest change may change the whole picture, it means you will never have to guess, but even beginning with guesses, you have to define a lot of the details. More details you have defined, harder it will be for your opponents to prove that one is wrong. :)

The answers given by Cort Ammon and Thucydides are exemplary in dealing with problems related to the propulsion system. This answer is more general, suggesting there are limits to the acceleration of any material object. particularly if that material object is a technological contrivance like a missile.

This limit is defined by the strength of materials composing the missile.

Strength of materials, also called mechanics of materials, is a subject which deals with the behavior of solid objects subject to stresses and strains .

In materials science, the strength of a material is its ability to withstand an applied load without failure. A load applied to a mechanical member will induce internal forces within the member called stresses when those forces are expressed on a unit basis. The stresses acting on the material cause deformation of the material in various manner. Deformation of the material is called strain when those deformations too are placed on a unit basis. The applied loads may be axial (tensile or compressive), or shear . The stresses and strains that develop within a mechanical member must be calculated in order to assess the load capacity of that member. This requires a complete description of the geometry of the member, its constraints, the loads applied to the member and the properties of the material of which the member is composed.

Taking into account the following materials failure modes, once acceleration exceeds any of these types of materials failures the vehicle under acceleration will itself fail.

Ultimate strength (tensile)

The maximum stress a material withstands when subjected to an applied load. Dividing the load at failure by the original cross sectional area determines the value.

Elastic limit

The point on the stress-strain curve beyond which the material permanently deforms after removing the load .

Yield strength

Point at which material exceeds the elastic limit and will not return to its origin shape or length if the stress is removed. This value is determined by evaluating a stress-strain diagram produced during a tensile test.

There will be an ultimate materials failure limit where all the materials making up the missile will be fractured, deformed and broken under its force of acceleration. Cort Ammon alluded to designing missiles so their components would form shapes or configurations where they would be strong enough withstand the adverse impact of accelerated stress.

To determine the specific failure loads applied by high acceleration it is necessary to know what materials are used in a given missile's construction. Once this is known, then it is possible to calculate the limiting acceleration of your missile.

• goof point, but - Once this is known, then it is possible to calculate the limiting acceleration of your missile. - not true, without knowing construction and how it works, you may even fail with determining upper limit of that acceleration. Commented Oct 22, 2016 at 2:18
• Ah yes, @MolbOrg, the point is I have failed to get my point clearly. The idea is not to determine the upper limit of acceleration exactly, but a ball park figure based on the point where the construction materials themselves will fail catastrophically. A good designer would aim at constructing a vessel capable of performing within viable operating parameters where its construction materials wouldn't fail. Commented Oct 22, 2016 at 4:32
• Yes, I understand your idea. Sure materials are significant part of design limitations, and when we talk about state of art, they are always part of limiting factors, as experience to use them and technology used in building that construction. If particular design is well known, and you change material, you may guess approximately what kind of improvement you might get from that, but if design is not known as OP case, there is really not much to say. As example is missile made from steel is possible - certainly; even 0.9c version of it? yheaa, kinda. Does that knowledge helps us? not so much. Commented Oct 22, 2016 at 6:22

Without knowing more specifics, you might be barking up the wrong tree. To accelerate quickly, you need the smallest and lightest mass possible. Rather than a "hunk of solid material" you wold be looking at a featherweight assembly of carbon nanotubes, graphine and aerogels. By analogy, consider the difference in acceleration between a modern Lotus sports car and a Toyota sedan. They are both powered by the same engines (Lotus buys their engines from Japan), but the engine is attached to a ridiculously light frame in the case of a Lotus, often weighting just a bit more than a motorcycle, resulting in insane performance compared to a Camrey.

The second problem on working this out is just what sort of "antimatter catalyzed fusion" rocket you are talking about? Deuterium fusion releases a lot of the energy in the form of neutrons, so a great deal of energy is being wasted , meaning your rocket will be much larger and heavier in order to generate the power output you want. The rocket could also vaporize under the intense heat unless there is an active cooling system.

Finally, just what are you shooting at exactly? what you are describing sounds almost like some sort of "point and shoot" system, in which case you might not be wanting a rocket anyway (the enemy will see the launch, and there is still a finite time to impact, allowing for countermeasures to be deployed). I can offer some real life devices which work on known physics which could seriously damage or destroy spacecraft, space stations or even surface installations on planets; specifically nuclear pumped weapons.

The Atomic Rockets conventional weapons page covers this in detail, so the summary is:

Nuclear explosions to drive pellets at speeds of up to 100km/sec

Nuclear "Shaped Charges" to drive a stream of liquid metal at up to 3% of c

The Casaba Howitzer to send a spear of star hot plasma at a target at 10% of c

Bomb pumped X-ray lasers delivering energy at the speed of light

Energy release is orders of magnitude faster than chemical reactions or accelerating missiles, so all you need is to have the warhead in range of the target (you could even dump them out the airlock like mines, if necessary). The RKKV you want would require orders of magnitude more size, mass and expense to accelerate to the velocity that could deliver the same amount of energy as a bomb pumped weapon.

Speed is your enemy in this situation. A ship can just move out of the way. A missile at that speed has no turning capability, its just a dart. A key point about light is it travels at C in every reference frame. So as the missile accelerates the target ship will see it and begin to move out of the way. At long distances even a .1 degree error in targeting angle becomes obnoxiously large. A relativistic projectile can be assumed to be visible to someone who is looking, and they have plenty of warning. At long ranges Doppler shift makes it obvious, all the light that touches it, bounces off at higher frequencies. Then you just move something in front of it to disperse the energy. At short ranges energy density constraints make going anywhere near C basically impossible, even with an antimatter engine. Masses of slower moving missiles, with say antimatter warheads, are a much more likely giant ship killer scenario.

I believe a4android's answer is exactly the information you're looking for. However, there may be a few things you haven't considered.

Any projectile at relativistic speeds is going to do world-destroying levels of damage. An explosive payload is not required. Specifically: A 10 gram bullet at .999c would have the energy of about 2*1018 J which is equivalent to a 480 megaton thermonuclear weapon. (I took this figure from physicsforums, I don't know if they accounted for special relativity and increasing mass.)

The only advantage that missiles would have is the ability to adjust their course once they've been launched. But again, at relativistic speeds this doesn't matter - a 2 degree change of course at near the speed of light would take half a solar system (and a TREMENDOUS amount of fuel).

What this means for you is that if you have a supercharged railgun capable of firing a bowling ball at .01c, it would ruin someone's day, somewhere and some time.

That is why we do not "Eyeball it".

• Just to nitpick the Mass Effect thing a bit, if you fire your magic railgun slug at .01c, it'll in all likelihood pass clean through the solar system into interstellar space and (a very long time later) into intergalactic space. It's really very unlikely to ruin anyone's day unless it actually hits the target or something very near it. Commented Oct 23, 2016 at 12:51
• The explosive payload was more to ensure the projectile doesn't cleanly penetrate smaller targets like ships and will instead fragment like a grenade and disperse the kinetic energy. Commented Oct 23, 2016 at 13:45
• @Elukka - Whether it takes a thousand years to reach another star system or a billion years to reach another galaxy, at some point that bowling ball will hit something. While space is mostly empty, it's not completely empty. On a large enough timescale, something will find itself on the receiving end of that boom. Commented Oct 23, 2016 at 18:07
• @Z.Schroeder - To reference another response on that PhysicsForums thread I linked to: "At such a high velocity, the kinetic energy of the various atoms and subatomic particles exceeds that needed to break them apart by quite a margin, and they will interact with the target as individual particles...the bullet will have the impact ballistics of a brief but extraordinarily intense burst of particle radiation." It's very likely that an explosive payload would not even function at those speeds, even if it could survive the acceleration without being crushed. Commented Oct 23, 2016 at 18:12