The Flying Manhole

[sevenperforce]

20 hours ago, RCgothic said:

https://en.m.wikipedia.org/wiki/Impact_depth

Newton's Impact depth approximation implies it'd struggle to escape the atmosphere.

It has to move through all space occupied by all the mass in its way, which must move faster than it to get out of the way.

But by conservation of momentum it can't give more velocity than it starts with to an equal or greater mass without coming to a complete halt.

The approximations look reasonable. Blunt body. High velocity. Non-cohesion of the impacted material.

The impact depth approximation provides a good rule of thumb for many cases but it is primarily applicable to high-speed impacts into soft but solid targets. For air impacts it is more of an order-of-magnitude estimate.

It wouldn't be hard at all for the cap to end up stabilized edge-on. The shockwave and shock heating effects would likely have an impact (no pun intended) on penetration dynamics.

[K^2]

Not touching whether it actually departed intact at 66km/s. Going to take that as a fiat just to see what would happen.

First, I would estimate how much speed is lost by the cover as it goes up. Absolutely worst case scenario at ludicrous speeds is if the object sweeps up all of the atmosphere. That's about 10T per square meter. In the mean time, the 4" steel plate is about 800kg per square meter. So the slowest the cover would be departing is at 18km/s. There is not enough matter between it and space to slow it down more. Notably, this applies to any fragments large enough that we can ignore viscous friction. At speeds as noted, just based on Reynolds numbers, it'd have to be in sub-mm sizes for that to be the case.

Given that, the cover has to survive only a fraction of a second. So we can throw away immense heat, friction at boundary, or any turbulence as possible sources of destruction. The only thing that can possibly do damage in this situation is the decelerating force being uneven, causing inertia to tear up the plate. Again, the maximum force can't possibly change. It's just impulse of air being swept by the cover. In the most extreme case, we can say that the force drops to zero at the edge. So lets take this now to be a disk 120cm in diameter that's sweeping 1.2kg/m³ of air at the center and experiences no drag at the circumference. So lets convert it into static weight.

The pressure at the center works out to a peak pressure P = 5.2GPa. Of course, we don't care about the absolute, but rather the gradient. About the worst case scenario here is if the pressure drops linearly to the edges. That will generate the most stress. Anything realistic will drop off sharper near the edge resulting in less stress. Take a ring of metal dr thick some distance r from center. It experiences aerodynamic force element:

dF = 2π P (1 - r/R) r dr

So total force up to some distance r:

F(r) = π P (r² - 2r³/(3R))

Notably, F(R) = (1/3) πR² P = ma = πR² d ρ a  for the disk of thickness d and density ρ. So we have net deceleration:

a = P / (3 d ρ)

Stress is then computed based on difference between aerodynamic and inertial forces.

ΔF(r) = F(r) - πr² d ρ a = (2/3) π P (r² - r³/R)

S(r) = ΔF(r) / (2πr d) = (1/3) P (r - r²/R) / d

And finally, S_max = S(R/2), so we can do the final analysis.

S_max = (1/6) P R/d

And since R/d for this disk is about 5.9, this actually works out very close to the same 5.2GPa of the peak aerodynamic force in the center of the disk. (It was always going to be a constant multiple of that based on geometry, but it being very close to 1 is just coincidental.)

This exceeds ultimate strength of steel by a factor ranging between "a few" and more than order of magnitude, depending on type of steel. Add to that the fact that most of this is actually going to be sheer, not tensile stress, and the fact that the cover is going to be heated up by the shockwave passing through it, and I don't think it has much chance under the worst case scenario.

 

So my money would be on the disk not remaining intact. At the same time, this is not something that tears the disk apart atom by atom. It's more like taking a compact disk and spinning it up to speeds where centrifugal force rips it apart, causing it to separate into multiple shards. Individual shards will have a far better chance of surviving the stress, as that R/d value is going to drop dramatically and air will be able to flow around the edges far easier. At the same time, there is still just so much matter between the shrapnel and space. That final speed of 18km/s isn't going to change unless shrapnel becomes small enough for viscous forces to take over, which would require very small fragments, on the order of no more than a few cm accross.

So if the disk wasn't pulverized by the shock, wasn't heated to a temperature where it's effectively a liquid, and did in fact gain 66km/s (which are all HUGE ifs,) I would bet on some amount of metal shrapnel a few cm in diameter to make it through the atmosphere heading out above the escape velocity. But almost certainly not an intact disk in any case.

[monophonic]

9 hours ago, K^2 said:

Not touching whether it actually departed intact

This leads in nicely to a though I have not seen discussed anywhere. Note I do not possess the math cells to arrive at a conclusion myself, so I am merely throwing this out here for anyone with the skill and interest to do so.

So the though goes, despite the concrete collimator between the device and the steel cap, the first thing from the explosion that reached the cap would have been the x-rays? So how strong would those x-rays have been? Could they have evaporated significant thickness of the plate in place? And how much force would that ablated steel cloud, expanding down the shaft, put on the steel cap as per Newton's third? How fast could it have pushed the now lighter remainder of the cap? Could that and the vaporized collimator cloud have combined for even more force?

Now I feel like a 5 year old kid asking questions far too fast for his parents to answer any of them... 

[kerbiloid]

These are remains of the Trinity test 30m high truss which survived a 20 kt explosion at 30 m above.

Spoiler

Pascal-B was 0.3 kt, at 150 m from the cap.

I.e. 70 times weaker, 5 times farther.

A 2 t heavy, 1.2 m wide concrete collimator would be (2 000 / 2 500) */ (pi * 1.2² / 4) ~= 0.7 m thick.

So, probably the cap would be not damaged before the mechanical pressure arrival.

[K^2]

34 minutes ago, monophonic said:

So the though goes, despite the concrete collimator between the device and the steel cap, the first thing from the explosion that reached the cap would have been the x-rays? So how strong would those x-rays have been? Could they have evaporated significant thickness of the plate in place? And how much force would that ablated steel cloud, expanding down the shaft, put on the steel cap as per Newton's third? How fast could it have pushed the now lighter remainder of the cap? Could that and the vaporized collimator cloud have combined for even more force?

Nothing X-Rays or evaporation of collimator do will have any impact on time scales involved. The only way a steel plate goes from rest to several (many?) km/s is due to arrival of the shock wave. It's the only thing that can give you these kinds of accelerations. And the way that works has nothing to do with shapes or material properties. The only thing that matters is density. A shock wave propagating in low density air hits high density steel. The shockwave is then reflected, and portion of the energy becomes kinetic energy of the plate as close to instant as mechanical interaction can get.

Now, what happens next does matter on how the shockwave hit the plate. If it's a nice planar wave hitting a nice flat surface, the plate will depart completely intact. If the wave is uneven, there's enough energy there to turn that plate into a fine spray. I don't know nearly enough about geometry of the shaft, nature of explosion, and any number of additional parameters to even guess which one it'd be closer to.

 

Btw, there seems to be a lot of misinformation flying about. I haven't found the original footage nor actual quote about velocity based on the footage. The 6x escape velocity was a rough estimate based on the shockwave propagation and was delivered before the footage was examined, and it's definitely an upper bound, not true velocity. The footage would place lower bound, and the actual velocity would be somewhere in between. I found a lot of citations claiming the cover was seen in one frame, but no other information that actually derives directly from that. Everyone seems to be conflating it with the 6x escape velocity upper bound.

[kerbiloid]

A shockwave is a wave of pressure, it doesn't throw objects, it compresses and shakes them.
A gas/air flow ("скоростной напор", can't find an English translation) does.

In air it's caused by the shockwave in air (passing through air , it heats it by local compression and makes to expand),
in vacuum it's an expanding cloud of the evaporated charge remains, 
in Orion it's caused by the expanding evaporated filler (that's why it should have low atomic mass) pushing out the evaporated membrane,
in this case it's a flow of evaporated collimator remains along the shaft.

Edited April 6, 2021 by kerbiloid

[monophonic]

Thanks @kerbiloid and @K^2 for putting that thing to rest.

28 minutes ago, K^2 said:

Now, what happens next does matter on how the shockwave hit the plate. If it's a nice planar wave hitting a nice flat surface, the plate will depart completely intact. If the wave is uneven, there's enough energy there to turn that plate into a fine spray. I don't know nearly enough about geometry of the shaft, nature of explosion, and any number of additional parameters to even guess which one it'd be closer to.

We have the claim that the plate was identifiably intact in that high speed film frame. If true that would point towards the relatively flat end of your scale. Curiously enough I seem to have a memory of seeing that film frame somewhere online once. Yet I cannot find it or any mention of other people seeing it anywhere. I must have dreamed it up or else it must have been an "artists impression" (i.e. a fake).

[K^2]

8 hours ago, monophonic said:

We have the claim that the plate was identifiably intact in that high speed film frame. If true that would point towards the relatively flat end of your scale. Curiously enough I seem to have a memory of seeing that film frame somewhere online once. Yet I cannot find it or any mention of other people seeing it anywhere. I must have dreamed it up or else it must have been an "artists impression" (i.e. a fake).

I also felt like I saw it somewhere, but yeah, maybe it was just an artistic rendering of what it would have looked like...

So the problem is that I can't imagine the picture would be clean enough for you to tell any features. I don't think you'd be able to tell an intact plate, from pile of shrapnel or effectively metallic spray that hasn't dispersed yet. So while yes, the fact that it looked plate-like rather than an expanding bubble suggests that the shock was mostly flat, which I'd expect from anything arriving via a long shaft, it's not enough for me to conclude that it was actually intact.

And again, size of shrapnel and velocity matter. If it was, indeed, on the high end of the estimate, in the 40-50km/s range, then unless the shrapnel was effectively a spray, these pieces would have punched through the atmosphere and only experience ill effects from that once in space departing Earth at escape velocity. In which case, I'd say launch counts, because nobody said it has to be an intact man-made object. But if the velocity was a lot lower or particle sizes really small, then the atmosphere would have created enough drag to slow these bits down enough to keep them here and, most likely, burn up.

[sevenperforce]

21 hours ago, K^2 said:

Given that, the cover has to survive only a fraction of a second. So we can throw away immense heat, friction at boundary, or any turbulence as possible sources of destruction. The only thing that can possibly do damage in this situation is the decelerating force being uneven, causing inertia to tear up the plate. Again, the maximum force can't possibly change. It's just impulse of air being swept by the cover. In the most extreme case, we can say that the force drops to zero at the edge. So lets take this now to be a disk 120cm in diameter that's sweeping 1.2kg/m³ of air at the center and experiences no drag at the circumference. So lets convert it into static weight.

The pressure at the center works out to a peak pressure P = 5.2GPa. Of course, we don't care about the absolute, but rather the gradient. About the worst case scenario here is if the pressure drops linearly to the edges. That will generate the most stress. Anything realistic will drop off sharper near the edge resulting in less stress. Take a ring of metal dr thick some distance r from center. It experiences aerodynamic force element:

This is exactly the analysis I was looking for.

**screams in joy**

But in thinking about this...how sure are you that the force has its maximum at the center and drops to zero at the edge? Why wouldn't it be the other way around?

Think about what we know from re-entry characteristics of a blunt body. Obviously we're talking about massively higher velocities here, but still. Maximum plasma temperatures are at the edge, not the center, because plasma temperature is a function of the compression ratio, and compression is highest at the edges. In the very center, the compressed air cannot escape and builds up, expanding both in the direction of travel and perpendicular to it, rarifying the air immediately in front of the vehicle. Thus the highest pressure would be on the edge, not the center.

Intuitively, we're still looking at a pressure gradient, not absolute pressure, so would it even make a difference? Well, there are two more considerations. First, the gradient will be distributed differently if the peak pressure is at the edge and decreases toward the center. More importantly, though, this pressure differential is certainly going to be more than what the rigidity of the steel can handle, and so it will turn the disc into a mushroom, almost like how the shockwave forces on a shaped charge mold it into a dart. The disc will be shaped more and more into a dart until the difference in forces is less than what its rigidity can handle, at which point it will punch through the remainder of the atmosphere like the tungsten telephone poles from Project Thor.

11 hours ago, K^2 said:

Btw, there seems to be a lot of misinformation flying about. I haven't found the original footage nor actual quote about velocity based on the footage. The 6x escape velocity was a rough estimate based on the shockwave propagation and was delivered before the footage was examined, and it's definitely an upper bound, not true velocity. The footage would place lower bound, and the actual velocity would be somewhere in between. I found a lot of citations claiming the cover was seen in one frame, but no other information that actually derives directly from that. Everyone seems to be conflating it with the 6x escape velocity upper bound.

The most recent interview with Brownlee seems to be this one with Tech Insider in 2016. They make it seem like Brownlee absolutely confirmed the tale and believes it left the atmosphere.

They also seem to indicate that the final speed estimate was done based on the footage.

[K^2]

1 hour ago, sevenperforce said:

But in thinking about this...how sure are you that the force has its maximum at the center and drops to zero at the edge?

Not even a little. But it's the worst case scenario, as there is more mass around the edges, and if the aerodynamic force is applied at the center, it's more likely to break the plate.

So all I am saying is that worst case scenario was enough to fracture the plate even if it was completely intact after the blast. Best case scenario is even load and the plate will easily depart the planet for good if it had at least 38km/s to start with. ( 11km/s * sqrt(10900kg / 900kg) )

There is still a question of whether the plate really was intact(ish) on that one frame, or if it was just a fine spray of shrapnel that hasn't had time to disperse yet. If the speed was, indeed, on the high end of the possible range and condition of plate was anything better than fine spray, I'm pretty sure at least some shrapnel was launched into interplanetary space. But I have no special confidence in either of these two conditions based on everything I was able to find. So *shrug*.

[kerbiloid]

7 hours ago, sevenperforce said:

The most recent interview with Brownlee seems to be this one with Tech Insider in 2016. They make it seem like Brownlee absolutely confirmed the tale and believes it left the atmosphere.

The faery tale lives its own life.

The fact that nobody since 1957 has calculated mv2/2 and still doesn't, preferring to blindly believe in someone's emotionally invented number based on such highly accurate measurement as "we saw it just in one frame, trust me", and completely ignoring elementary calculations makes me feeling so sorry about the future of scientific science...

It's probably too hard to accept that Sputnik was the first extraterrestrial object since the dinokiller ejecta.

Edited April 7, 2021 by kerbiloid

[RCgothic]

I'm pretty sure this is the sort of thing we could sim these days.

[sevenperforce]

15 hours ago, K^2 said:

Not even a little. But it's the worst case scenario, as there is more mass around the edges, and if the aerodynamic force is applied at the center, it's more likely to break the plate.

So all I am saying is that worst case scenario was enough to fracture the plate even if it was completely intact after the blast. Best case scenario is even load and the plate will easily depart the planet for good if it had at least 38km/s to start with. ( 11km/s * sqrt(10900kg / 900kg) )

Oooooh, I see. I thought you were trying to do a best-case scenario, not a worst-case scenario. Missed that part.

I guess we just need Elon to buy a Russian nuke and test it.

[SOXBLOX]

1 hour ago, sevenperforce said:

I guess we just need Elon to buy a Russian nuke and test it.

That...sounds like something he'd try.

[wumpus]

2 hours ago, SOXBLOX said:

That...sounds like something he'd try.

The real question is what happens when they don't sell him a nuke.  What does he come up with?  Is it safer for the US or Chinese just to sell him a nuke?

[K^2]

3 hours ago, sevenperforce said:

I guess we just need Elon to buy a Russian nuke and test it.

If that 300T yield thing is correct, you can do this with HE. Just set up detonator at the bottom, so the shock accelerates through secondary as it goes up the shaft. The shaft is tall enough that by the time the shock arrives at the top, there shouldn't be any substantial difference between conventional and nuclear explosion at that yield.

That's still a lot of explosives, and I don't know if a demo company would be allowed to handle that much, but it still ought to be easier to get than a nuke, right? Right? Please, please be right.

[sevenperforce]

30 minutes ago, K^2 said:

If that 300T yield thing is correct, you can do this with HE. Just set up detonator at the bottom, so the shock accelerates through secondary as it goes up the shaft. The shaft is tall enough that by the time the shock arrives at the top, there shouldn't be any substantial difference between conventional and nuclear explosion at that yield.

That's still a lot of explosives, and I don't know if a demo company would be allowed to handle that much, but it still ought to be easier to get than a nuke, right? Right? Please, please be right.

The primary limitation on the sale of nuclear weapons is the Treaty on the Non-Proliferation of Nuclear Weapons (NPT), which absolutely prohibits nuclear powers from "transfer[ing] to any recipient whatsoever nuclear weapons or other nuclear explosive devices." 

However, Article V of the treaty permits "peaceful applications of nuclear explosions" to be "made available to non-nuclear-weapon States" which have ratified the treaty provided that there is "appropriate international observation" and that "the charge to such Parties for the explosive devices used will be as low as possible." 

The trouble, presumably, is that Elon Musk is not a "State" under the purposes of the NPT and cannot sign the treaty. Nuclear-weapon states are not permitted to make nuclear weapons available to peaceful purposes to non-nuclear-weapon states which have not ratified the NPT. So unless Elon could be defined as a state then he can't buy one.

However, if Elon can't buy a nuke outright, could he pay the US or Great Britain or Russia or China to put one in a specifically-designed test chamber and trigger it?

The Comprehensive Nuclear-Test-Ban Treaty has not yet come into force, but if it ever does, then the prohibition on "causing, encouraging, or in any way participating in the carrying out of any nuclear weapon test explosion or any other nuclear explosion" would also be a bar.

The 1963 Limited Test Ban Treaty requires its signatories to "prohibit, to prevent, and not carry out" any nuclear explosion, including peaceful explosions, in the atmosphere, in outer space, or in water. It further prohibits nuclear explosions which would cause "radioactive debris to be present outside the territorial limits of the State under whose jurisdiction or control such explosion is conducted." 

So there's no prohibition on releasing radioactive debris; you just have to make sure said radioactive debris remains inside your territorial limits. But presumably, blowing up a nuke at the bottom of a tube in order to yeet a manhole cover out of Earth's gravity well would tend to send at least some debris high enough that it would drift outside of territorial limits. 

[magnemoe]

2 hours ago, K^2 said:

If that 300T yield thing is correct, you can do this with HE. Just set up detonator at the bottom, so the shock accelerates through secondary as it goes up the shaft. The shaft is tall enough that by the time the shock arrives at the top, there shouldn't be any substantial difference between conventional and nuclear explosion at that yield.

That's still a lot of explosives, and I don't know if a demo company would be allowed to handle that much, but it still ought to be easier to get than a nuke, right? Right? Please, please be right.

Some mining companies have used over hundred tons in some of their more spectacular blasts. 
Demolition tend to not require so much explosives and bringing down an mountain side. 
Has also been some kiloton tests with piles of high explosives to simulate nuclear blasts but this has been military tests.

Now its an problem here using high explosives, cannons has an maximum velocity, granted you are using high explosives not cordite and your barrel is 10 times longer than the Paris gun but the shock wave still have an speed limit. With an nuclear bomb you heat up the concrete plug to very high temperature plasma who will move much faster. 

[kerbiloid]

There is a country where the impossible can get possible, but I won't tell to avoid politics.

[K^2]

3 hours ago, sevenperforce said:

The trouble, presumably, is that Elon Musk is not a "State" under the purposes of the NPT and cannot sign the treaty.

[...]

However, if Elon can't buy a nuke outright, could he pay the US or Great Britain or Russia or China to put one in a specifically-designed test chamber and trigger it?

[...]

So there's no prohibition on releasing radioactive debris; you just have to make sure said radioactive debris remains inside your territorial limits. 

I really appreciate that you clearly did research on this. This almost reads like a legal opinion.

3 hours ago, sevenperforce said:

But presumably, blowing up a nuke at the bottom of a tube in order to yeet a manhole cover out of Earth's gravity well would tend to send at least some debris high enough that it would drift outside of territorial limits. 

So this one might be sticky, but my (admittedly limited) understanding is that space is only outside your nation's borders if you are a signatory to Outer Space Treaty. Quick lookup shows that there are nations that are signatories to NPT, but not OST. These nations can claim that launching a manhole cover vertically up doesn't cause it to leave their borders, while at the same time have the right to request a nuclear detonation on their territory subject to NPT.

Notable candidates would include Turkmenistan, Uzbekistan, Tajikistan, and Kyrgyzstan. I'm guessing because they haven't bothered to sign OST following collapse of USSR.

Of course, any nation that could provide a nuclear device under NPT is also a signatory to OST. So I don't know if they would apply their definition of nation's borders and it would then be considered a violation of NPT from their perspective? Or is it only the laws and treaties of the nation where the test would be conducted that apply? Thoughts?

2 hours ago, magnemoe said:

Some mining companies have used over hundred tons in some of their more spectacular blasts. 

Mining companies tend to use slow explosives, though. These are usually tertiary explosives that are a bit easier to work with in large quantities.

So yeah, finding anyone but military that can conduct such an explosion might be problematic.

[monophonic]

Some legal issues might be side stepped if you mix your explosive components in the chamber itself just before the detonation. Now tell me how much stoichiometric methalox can you pump into a chamber before it spontaneously ignites? Presumably the methane and oxygen should come from separate inlets but we would also want them to evenly mix before setting them off so probably not at the opposite sides of the chamber.

[sevenperforce]

On 4/7/2021 at 8:01 PM, K^2 said:

I really appreciate that you clearly did research on this. This almost reads like a legal opinion.

I appreciate that. I may or may not be in law school, so........

On 4/7/2021 at 8:01 PM, K^2 said:

So this one might be sticky, but my (admittedly limited) understanding is that space is only outside your nation's borders if you are a signatory to Outer Space Treaty. Quick lookup shows that there are nations that are signatories to NPT, but not OST. These nations can claim that launching a manhole cover vertically up doesn't cause it to leave their borders, while at the same time have the right to request a nuclear detonation on their territory subject to NPT.

The OST doesn't prohibit the release of radioactive material into space; it just prohibits, substantively, (a) the placement of "nuclear weapons or other weapons of mass destruction in orbit" and says that member States (b) "shall avoid harmful contamination of space and celestial bodies" (emphasis mine).

So the OST doesn't prohibit the atmospheric or exoatmospheric release of nuclear byproducts as long as no nuclear weapon was placed in orbit and any associated contamination is harmless (which this would presumably be).

The "territorial limits" verbiage in the 1963 Limited Test Ban Treaty is the problem. There have been several instances in which the US or Russia (or, formerly, U.S.S.R.) performed underground tests which accidentally released radioactive byproducts into the atmosphere, which subsequently crossed international boundaries in minute quantities. In each case the opposite side complained bitterly but nothing was done, presumably because it was accidental. It's unclear whether the deliberate release of radioactive byproducts from a nuclear test (however peaceful) would be treated similarly.

Although I suppose the signatories could readily enough ratify a codicil specifically excusing a one-time peaceful nuclear test, financed by Elon, under international supervision. It's not like the U.N. has any enforcement capabilities anyway. But I think the NPT would still prohibit Elon from purchasing a nuke outright.

[K^2]

2 hours ago, sevenperforce said:

The "territorial limits" verbiage in the 1963 Limited Test Ban Treaty is the problem.

My argument is that countries that aren't signatories to OST don't recognize space as being beyond their territorial limits, nor the fact that they cannot lay territorial claims on anything in space. Such nations can easily claim the debris as the extension of their territory. Declare it a consulate or a vessel in international waters if it helps. It's paper thin, sure, but if there is no reason for any other nation to specifically challenge it, then maybe it's good enough.