There has been a discussion going on on the Star Frontiers Discord server (invite link) in the #KnightHawks channel about firing one of the Knight Hawk (KH) rocket weapons (torpedoes – TT, assault rockets – AR, or rocket batteries – RB) from a ship downed on a planet surface at a target in orbit around the planet.

Given the nature of the weapons, my gut reaction is that they have more than enough power to get from ground to space, but I want to check that assumption. There are other things to consider such as aiming, steering, etc. Let’s look at these in turn.

Power to get to orbit

The first question we have to answer is if the weapons have enough power to get from the ground to orbit. They are designed for firing in a zero gravity environment and getting to a target in orbit means that they have to climb out of the gravity well. Can they?

We start by looking at how much thrust the weapons have. And to do that, we have to make an assumption about how their engines work. We know that they can cover either 40,000km (TT & AR) or 30,000km (RB) in one KH turn or 10 minutes. The question is do they constantly accelerate over that time or is there a short burst of acceleration and then they basically coast? Personally, I’ve always felt that torpedoes do the former while AR and RB do the latter. But let’s look at the accelerations involved. Warning, math ahead!

The equation that governs the distance traveled for an object that is accelerating under thrust for a time and then (possibly) coasting is given by

d(t) = \frac{1}{2}at_{1}^2 + v_{max}(t-t_1)

where

• d(t) is the distance traveled at time t >= t₁
• t is the total time of the calculation
• t₁ is the time when the acceleration stops
• a is the acceleration of the object while under thrust (assumed to be constant here)
• v_max is the velocity of the object after acceleration is over and is simply the acceleration times t₁. Substituting this in gives us

d(t) = \frac{1}{2}at_{1}^2 + at_1(t-t_1)

Simplifying this slightly gives us

d(t) = att_1 - \frac{1}{2}at_{1}^2

This equation can be used in both cases where the object accelerates and coasts (t>t₁) and when it accelerates the entire time (t = t₁).

For the KH weapons, we know the total distance d (30,000 or 40,000 km) and the total time t (10 minutes or 600 seconds). What we don’t know is a or t₁. But we can pick one and calculate the other. The rocket with the highest acceleration on Earth is the Sprint anti-ballistic missile which had an acceleration of 100g (982 m/s²) for 5 seconds. So as long as we stay below that figure, we are probably safe.

For the accelerate constantly option, which is what I assume torpedoes do, to cover 40,000km in 10 minutes, the acceleration would need to be 222.22 m/s². If a rocket battery had the same acceleration pattern, the acceleration would only be 166.67 m/s² as it has to cover less distance in the same amount of time.

A reader on Facebook asked about the decrease in range and when looking into that, and adding the section at the end, I realized I had miscalculated the acceleration in the burn and coast mode. The actual numbers (now corrected) were lower than I first posted and much more reasonable.

Now for the accelerate and drift option (which I assume for AR & RB), let’s start by trying a 30 second acceleration and then coast to the target. To achieve that, the AR would need an acceleration of 2279.20 m/s² and the RB would need an acceleration of 1709.4 m/s². Which is starting to be a bit excessive and well over the 100g of the Sprint rocket. If we go with a 60 second acceleration period, those drop to 1169.59 m/s² and 877.19 m/s² respectively. Those are more reasonable and given advanced materials, and the fact that these are fired in a vacuum, maybe that is reasonable.

So now we need to see if these accelerations (and durations) are enough to get these missiles off of a planet.

When we launch from the surface of the planet there are three forces we need to deal with. The first is the thrust from the rocket which is just mass of the rocket times the acceleration we just calculated. This pushes the rocket up. The second is force of gravity pulling the rocket back, and finally we have drag on the rocket as it pushes through the are. This also resists the rocket getting off the planet.

The drag force is given by:

F_d = \frac{1}{2}C_d\rho Av^2

where

• C_d is the drag coefficient
• rho (ρ) is the fluid density of air
• A is the cross sectional area of the missile
• v is the velocity of the missile

The drag coefficient depends on the shape of the object and how the fluid flows around it. I’m just going to use the drag coefficient for a long cylinder (0.82) as an approximation for any of these missiles. That value comes from the Wikipedia page for drag coefficients.

The other bit we need is the fluid density of air. That can be approximated by

\rho(h) = \rho_0e^{-h/h_0}

where

• ρ₀ is the fluid density of air at sea level (we’ll use 1.222 kg/m³)
• h₀ is the scale height of the atmosphere (10.4 km).
• h is the altitude

If you want to read more about the exact calculations, check out the Density of Air Wikipedia page.

To figure out the acceleration as a function of time, we add up all those forces and divide by the mass of the rocket. That acceleration can then be used to find how high the rocket goes. The problem is that the acceleration has terms that depend on the velocity and position which in turn depend on the previous acceleration and velocity. This type of problem is known as a partial differential equation. And it can be solved but is definitely beyond the scope of this blog. Luckily, it easy to approximate this one with a simple numerical iteration that you can do with a program or even a spreadsheet. So that’s what I did.

The only thing we don’t have is the mass of the rockets. It cancels out for the rocket thrust and gravity but we still need it for the drag term. The problem is that all the rules give us is the volume for each weapon. Since a larger mass will reduce the effect of the drag force, we’ll approximate on the light side and say that the missiles weigh only 1000 kg/m³, the same density as water. That’s might be light but I also assume that the volumes given in the rules are for the entire storage space for the missile, not just the missile, so it probably evens out. And that just means that if it is more massive, the drag effect will be smaller and easier for the missile to get to orbit. That gives us a mass of 10000 kg for the RB and AR and 20000 kg for the torpedo. Although for the rocket battery, I’m going to use a mass of only 1000 kg because it’s a salvo of smaller rockets.

Let’s start with the with the continuous acceleration engine. We’ll look at the RB, AR, and TT and see how long it takes to get to space (if it can at all). For the purpose of this calculation, I’m defining reaching space to mean it got to an altitude of 600 km. That’s a bit higher than the orbit of the Hubble Space Telescope. I’m also going to round the acceleration of gravity up to 10 m/s² and ignore the fact that gravity drops off with altitude. Finally we’ll use a cross section of one square meter in the drag equation (that’s probably high for the RB but that just means that any values we get for the RB are conservative). Here’s what we get using 0.25 second time steps on the integration:

The small mass of the individual rocket battery rockets definitely has an effect on the time. Even with the same acceleration as the other weapons, it would take 94.75 seconds to get to space. Still though, it takes less than 2 minutes to reach space and in truth the drag of the atmosphere is negligible for the RB after 57 seconds and 28.5 seconds for the AR. And remember, these are slightly longer times than the actual values because we’re ignoring the dropping off of gravity with height.

Now let’s look to the burst acceleration case. We’ll use the 60 second burn case. This gives us the following values:

Now there would be variations due to atmospheric density and gravity but even increasing the gravity to 1.5 g only added 0.75 seconds to the AR in the continuous acceleration case. Doubling the air density and keeping the gravity at 1.5g pushed the AR’s time-to-space up to 82.5 seconds. The thrust of these weapons is just so large that gravity and drag are only small effects.

In all cases, the missile reaches space before the engine cuts off. The drag force gets really high on these later cases because they are accelerating quickly but it’s not enough to keep them from reaching orbit. So it’s pretty safe to say that any of the weapons have the thrust to reach a ship in low (or even high) orbit if fired from the planet’s surface. But having the thrust to do so isn’t everything.

Heat of Passage

These missiles are going to be screaming through the lower atmosphere for the first 30-60 seconds and that drag force is going to generate a lot of frictional heat on the surface of the missile. The Swift anti-ballistic missile dealt with this by having an ablative heat shield to protect it through the lower atmosphere. It’s safe to assume that our missiles don’t have that as they were never designed for atmospheric launch but rather were expected to be working in a vacuum. One might argue that they might not even have a shell or skin, just structural supports to hold all the pieces as no aerodynamic shape is needed, but we’ll assume that just for the sake of handling they do have a solid surface.

This means that while there might be enough thrust, it might be too much for the missile to handle. The rocket battery, with the slower thrust, goes a bit slower so the effect isn’t as great but it spends more time hot. This is probably enough to make this a no go but it’s up to you to decide if this is a factor for your game.

Aiming/Guidance

The next consideration is how to aim and/or guide the missiles at their targets. And this comes down to how you feel the weapon is fired in space. This will impact if a ship sitting on the surface of the planet can fire and guide the weapon to its target. It really comes down to two ideas, direct flight or guided flight. Direct flight is where the missile is launched and it just flies in a straight line at the target based on its launched direction. Guided flight is where the missile has maneuvering capabilities and can steer itself after the target. Here’s my take on each of the three weapons we’re considering.

Assault rockets are moving player only and forward firing. These are mounted in a launch tube aligned with the main axis of the ship and the ship needs to be lined up with the target before firing as the assault rockets are direct fire. Once launched, they fly in a straight line at the target and detonate. If they have any steering capability, it is minimal for final approach.

Like assault rockets, rocket batteries are direct flight. The difference is that the launcher itself can swivel to aim the rockets and control the direction of launch. Again, any guidance of the missiles themselves if very, very minimal, probably confined to final approach.

Torpedoes, on the other hand, I treat as a guided flight weapon, probably by an internal system rather than controlled by the firing ship, but they do have the ability to steer to chase their targets. You fire them out of the side of the ship and they steer around and guide in as they approach.

This has an impact on how these weapons navigate through the atmosphere. Since they are designed for spaceflight, and not to work in an atmosphere, it’s very unlikely that any of these missiles have fins, which would help to stabilize their flight though the air. They are not needed in space so the missiles wouldn’t have them. Which means, that for the direct flight weapons, strong winds, turbulence, and uneven air flow are going to push them off their intended path, possibly significantly and maybe catastrophically as they don’t really have any guidance system to keep them on track. This won’t be so bad for a guided weapon as it has the capability to somewhat steer itself and remain on course.

Another question is rifling of the launch system. This is where the launch system either by physical design or the way the rocket is made, imparts a spin to the missile as it is launched. While not needed, it may very well be a design feature of the AR and RB launchers as it imparts stability to the flight and could help keep the missile from going off course due to minor unevenness of engine thrust over the long flight, basically a form of gyroscopic stabilization. If you do include some sort of spin stabilization in your launch system, that could help the missile stay on course while buffeted by the atmosphere.

Given those aspects of the missiles, you’d only be able to launch an assault rocket from the surface of the planet if the ship had landed tail down and the target passed nearly directly overhead. Since it takes the whole ship to aim the weapon, a grounded ship can only shoot and AR at a target that passes directly in front of it. The rocket battery or torpedo launcher could conceivably fire at any target. Additionally, there is a good chance that the AR or RB will get knocked off course due to the turbulence of passage through the atmosphere whereas the torpedo, with its steering capability, might fare much better.

Environmental impact

Here I’m not talking about whether the missile launch is good for the environment or not (it’s probably not) but rather what direct impact it has on the firing site. The ship itself would probably be fine as it was designed to handle the launch of the weapon. But they were designed for space and now you have all this heat being generated by the rocket launch rushing out into the surroundings. The air will get hot, things might catch on fire, etc.

Additionally, even the “slow-moving” rocket battery breaks the sound barrier in 2.5 seconds while it is only half a kilometer away from the ship. What impact does that sonic boom have on the ship and surroundings? What impact does it have on the missile?

Effect on range

I had originally ignored this because I was assuming we were looking at shooting at something in orbit around the planet which is well within the range of all these weapons. A commenter in the STar Frontiers Facebook group asked about it so I thought I’d post the max ranges for each of the weapons in each scenario.

If you assume an unguided missile (i.e. AR or RB in my case), I’d definitely reduce the chance to hit something not in orbit just because of the aiming issue, the atmospheric passage would reduce the accuracy of the weapon. But here are the max ranges for the weapons (these are for vertical launches):

So for the larger weapons, the atmosphere only reduces the range by less than 3,000 km or less than 10%. The RB fares a little worse due to the increased drag. In continuous burn mode, it loses ~4,300 km or 14.4%. The biggest effect is when you assume the RB is a burn and coast engine. Then the effect is nearly 8,000 km off it’s normal range, a 26% reduction.

Obviously firing at an angle through more atmosphere would reduce this even more, firing the continuous burn RB at a 30 degree angle drops the range to ~22,000 km and the AR to ~32,600 km as examples. Also, these are actually slight underestimates of the range because I didn’t taper off the gravitational effect with distance from the planet. But that is such a small effect relative to the thrust that the variations are minor.

Final thoughts

This was a fun problem to think about. I think in my game, I’d allow these weapons to be fired at targets in space but give them a failure chance to go off course (possibly catastrophically) during their flight through the atmosphere, probably 20% for torpedoes, 40% for assault rockets, and 60% for rocket batteries. If the PCs were doing this, I’d reduce those failure changes for various things that they do to mitigate some of the problems mentioned above.

Such mitigations could include reprogramming the rockets for slower initial launch speeds (assuming you declare to rockets to not have a solid fuel core that cannot be regulated), somehow inducing spin stabilization at launch, adding fins that deploy after launch to stabilize flight, etc. Of course those things would probably take hours of work to set up and modify the missile so it wouldn’t be something they could do quickly.

One thing I didn’t talk about was the extra time it would take to get out of the atmosphere if not launching straight up. The launch angle (theta), measured in degrees from vertical, is in my spreadsheet that does the calculations but I only presented the numbers for straight up. Also, that angle only affects the height calculation and isn’t accounted for in the slow down caused by gravity but it’s a relatively minor effect unless theta is large. As a first approximation, just divide the times listed above by cos(theta) to get the time to space for a different angle. It’s actually a little bit longer than that due to more time spent with the drag force affecting the missile but not by much (at least for small theta). I’d probably also increase the failure chance by the same factor.

Speaking of the spreadsheet, you can find it here if you want to play with it:

What thoughts do you have about this scenario? Share your ideas in the comments below.