Could any existing gun reach the Karman Line?

There's a running Kickstarter to build a HARP-like gun that is multi-chambered and can fire a payload into "space". I don't see where this person's criteria for space is, except for the fact that it's intended to be suborbital. I would assume this means that the payload gets over 100 km high. As I understand it, HARP did not fire up to the Karman line.

https://www.kickstarter.com/projects/1682852725/the-starfire-space-cannon

Thinking about the numbers, the minimum delta v should be determined by high school level kinematics:

$$h = v_0 t - \frac{1}{2} g t^2 \\ v = 0 = v_0 - g t \\ v = \sqrt{ 2 h g } \approx 1.4 \frac{km}{s}$$

While this would be difficult to achieve with a single chamber, it is on the very very far end of common muzzle velocities. Searching around for it, tank guns look like they can get 1,750 m/sec. Of course, I've left out the hard part - air resistance.

Are there any existing weapons that, if aimed upward, would reach the Karman line? And are there any documented examples of this?

• In the kickstarter, I believe the guy even uses the term "orbit" at one point, which is when I gave up hope on the project. Either he is mistaken in his terminology or completely over the top ambitious in trying to fire a projectile into orbit. That said, I'd be happy to have him prove me wrong a leave a little projectile in orbit!! Seems highly unlikely.. – Steve Midgley Feb 12 '14 at 2:41
• @SteveMidgley I'd really like to hear his strategy for making the circularization burn that would be required to make orbit... – Erik Feb 12 '14 at 3:42
• Related question Can gunpowder get you to the moon? – James Jenkins Feb 12 '14 at 11:16
• @Erik It is not orbital. The parts where he says that, he's talking about what he would like to eventually do someday. The Kickstarter is limited to suborbital. – AlanSE Feb 12 '14 at 12:34

Yes, it's been done:

In the Project HARP a U.S. Navy 16 in (410 mm) 100 caliber gun was used to fire a 180 kg (400 lb) slug at 3600 m/s or 12,960 km/h (8,050 mph), reaching an apogee of 180 km (110 mi), hence performing a suborbital spaceflight.

Or, if you follow this link from the Kickstarter campaign you mention (Graf's excellent history of the HARP project):

The third and final 16-inch gun of the HARP program was installed at the Yuma proving grounds in Arizona. . . . . The Yuma Gun's sole claim to fame was that on November 18, 1966 it lofted a Martlet 2 vehicle to a world record altitude of 180 km, which still stands today.

• I'm a bit surprised you didn't mention your answer on Is there any non-rocket spacelaunch concept within reach of current material science and technology? John Hunter's concept seems a lot more feasible to me than the Starfire project and involves orbit insertion burn for what's initially a ballistic projectile. – TildalWave Feb 12 '14 at 9:04
• I was initially highly skeptical of Richard Graf's project myself, but less so after reading his article on Gerald Bull (a case study in the triumph of politics over science). If you drill down through the comments at the end of the Kickstarter page, you'll see he's reduced the initial G-force to 10,000 Gs with his approach, still way over the top for squishy humans, but sufficient (he claims) to leave properly-designed electronic components undamaged. – Jerard Puckett Feb 12 '14 at 14:47
• Proximity and radar electronics are routinely used in AAA gun projectiles that produce 30,000 Gs. – C. Towne Springer Feb 16 '14 at 17:57

I was hoping that I could say this is physically feasible, but after running the numbers, it unfortunately seems not. Let me start by listing the various claims on the Kickstarter:

• Bore diameter: 8 inches
• Maximum pressure: 60,000 psi
• Tube length: 45 feet
• Altitude: 274.9 m above sea-level
• Muzzle velocity: 1500 m/s (mentioned in comments, I may relax this later)
• Maximum altitude of the payload is taken to be 100 km by the "space" qualifier

The bore diameter kills it. You can't do it with these parameters. I'll show this by establishing a maximum and minimum mass for the shot based on obvious physical constraints. I will show that these requirements conflict, and can not be satisfied by this device.

Maximum mass

This is simple, because we're given a maximum pressure from combustion. The core concept of the project is that it will be multi-chambered. All this does is allow them to sustain that driving pressure over a longer segment of the barrel than a single combustion gun would allow. So the actual pressures might be lower than this, but they can't be higher. This easily constrains the kinetic energy that the projectile can have. The calculation below is simple, but this only introduces conservatism. I'm not introducing back-pressure from the air or anything - so I'm being as generous as possible here.

$$E = F \times d = \left( P A \right) \times d = (60,000 \text{ psi})\pi(4\text{ in})^2(45 \text{ ft}) = 184 MJ$$

This seems simple enough. If we take the muzzle velocity to be 1500 m/s, then we find that the maximum mass that can be shot is 163.5 kg. The payload must be lighter than this, or else the gun can not accelerate it to the required speed.

Minimum mass

In the OP I gave the required velocity needed to make it to the Karman line in the absence of air resistance. Note that the currently quoted muzzle velocity is 100 m/s higher than this. Earth's atmosphere is clustered mostly around the first 10 km of sea-level. So one approximation we can make it to imagine that atmosphere as a thin sheet that the bullet must make it through, in other words, we don't credit the loss of speed due to gravitational potential along this track.

If we take the mass-thickness of the atmosphere to be $\mu$, we can write the energy lost to drag as follows:

$$\Delta E =\frac{1}{2} \mu v^2 A C_d = 150.2 MJ$$

A generous assumption I made was to take the $v$ in this equation to be the velocity needed to reach the Karman line, 1.4 km/s. That's the lower of the options, so this will underestimate the impact of drag. The area (again) comes from the bore diameter. The drag coefficient can be looked up. I'll take it to be a perfect sphere, with a coefficient of 0.47.

This is only the energy lost to drag. The projectile has to get past this and then still have enough energy to make it to the Karman line. This is straightforward to set up mathematically. With this, we establish a minimum mass.

$$\frac{1}{2} m v_1^2 - \Delta E = \frac{1}{2} m v_2^2 \\ m = \frac{ \Delta E }{ \frac{1}{2} ( v_1^2 - v_2^2 ) } = 1,035.9 kg$$

Projectile must be more massive than this in order to preserve its momentum as it makes it past the atmosphere. The contradiction is obvious. It needs to be this massive, but the gun can't supply the energy in order to get it there because the barrel simply isn't large enough.

You can't fix this by tweaking the parameters either. In the above equation, I used 1500 and 1400 m/s for the v1 and v2. But make v1 larger. It doesn't reach parity with the other requirement until well over 2,000 m/s. We all agree that the kinetic energy of the explosives aren't enough the achieve this. It's outright physically impossible. See many other questions on this site about the subject for elaboration to why. This is why HARP moved on to light gas guns - to get closer to orbital velocities.

Compared to others

The tank gun example I cited also fails. Its parameters come out even worse than the Starfire. With the possible exception some heavy munitions of battleships, the answer to the question I asked is more or less "no".

The accepted answer has detail on the HARP firing that did what this project is trying to do. So it's possible, but there is a critical difference - they used a 16 inch bore. The theme of my calculations is that an 8 inch bore is too small. If you double the bore diameter, you quadruple the maximum energy you can put into it. I believe it was also longer, which puts it even further away from the energetic constraint here.

It looks like the Starfire project might be able to launch the payloads a few kilometers up, but it can't go further than this. It can't physically happen. I'm eager to hear any challenges to the math I've used and the assumptions therein. But as it looks now, this isn't getting anywhere close to space.

• This answer assumes that the projectile must be the same diameter as the barrel. Tanks commonly fire 'Discarding Sabot' rounds, where a long thin heavy core is surrounded by a light block that falls apart as it leaves the barrel. Otherwise, it's an excellently detailed answer! – Robin Bennett Jun 4 '19 at 9:49