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The second stage of Apollo 11 generated a greater than 20 deg shock-wave half angle which corresponds to less than $1000 \frac{m}{s}$ forward speed not 6000 mph ($2682.24 \frac{m}{s}$)?

I measured the half angle of the shock-wave generated by the second stage of Apollo 11 after separation (see the picture). This angle is greater than 20 deg.

Using the relation that gives the velocity of an object as a function of the speed of sound and the half angle of the shock-wave, I got:

${{V}_{Stage\ 2}}<\frac{1225\frac{km}{h}}{\sin \left( 20{}^\circ \right)}=994.90\frac{m}{s}$

According to NASA, at burnout, the first stage had 6000 mph.

In appears that, after separation, stage II advanced only at 994.9 m/s / 6000 mph = 37.09% of the stated speed (6000 mph). How is it possible?

First Stage Separation, Apollo 11

Source: Video, First Stage Separation, Apollo 11

A more clear picture of the same event.

First Stage Separation, Apollo 11

Source: Saturn V

Update

Another picture, from the same video, shows the shock-wave created by the leading edge of Saturn V. It is visible due to the cloud of gas formed around the entire rocket when the first stage retrorockets fired during separation. The same half angle, greater than 20 degree, appears.

enter image description here

Source: Video, First Stage Separation, Apollo 11

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    $\begingroup$ Did you account for changes in the speed of sound at altitude? $\endgroup$ – kim holder Aug 13 '16 at 19:08
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    $\begingroup$ I assume we see the rocket not from the side but at some angle. This makes the angle you measure larger than it actually is. $\endgroup$ – asdfex Aug 13 '16 at 19:16
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    $\begingroup$ What in your image did you measure? can you put some lines and the angle on it? I can not see any visible shockwave in that image. There is the exhaust plume, but thats about it. If you watch a launch video, the shockwave is visible much earlier in the flight, both on the petal adapter and the adapter connecting the S-II with the S-IVB. $\endgroup$ – Polygnome Aug 13 '16 at 22:31
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    $\begingroup$ @kimholder &RobertWerner - the speed of sound (bulk) is a reflection of the average thermal speed of the molecules themselves, so to first order and at atmospheric type pressures, it's proportional to the square root of temperature (in Kelvin) but mostly independent of pressure.I have to work hard to remember to remind myself of that every time. $\endgroup$ – uhoh Aug 14 '16 at 3:37
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    $\begingroup$ I can't believe Phil forgot to touch up the shockwave videos to show the "correct" angle and that's what finally brought down the conspiracy after all this time. My life is over. $\endgroup$ – Russell Borogove Dec 2 '16 at 3:01
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That is not a shockwave. It looks like the edge of the second stage engine exhaust plume.

If it were a shockwave, more shockwaves should be visible: one at each point where the diameter of the rocket changes, so escape tower, CM, both interstages.

Shockwaves are rarely visible. In the lower atmosphere you can sometimes see condensation in the low-pressure area aft of the shockwave (a vapor cone). You normally need a Schlieren camera to see the shockwave itself.

I've reviewed a number of launch videos. No shockwaves visible anywhere. The vapor cone is visible around Mach 1, not at higher speeds.

Stage separation occurred at 67 km, atmospheric pressure at that altitude is very low. I don't think there's enough air (or water in the air) to create a visible shockwave.

Here's the separation sequence:

Update: Here are some images of actual shockwaves. This is a shadowgraph of a bullet. A shadowgraph is a way to photograph objects that makes shockwaves more visible.

enter image description here

So we see a prominent shockwave at the tip of the bullet, plus smaller ones aft which are at a different angle. This means you can't reliably measure speed from the angle of the shockwave alone. You can also see the turbulent wake region aft of the bullet. This region has the lowest pressure, and will be the first to be filled with exhaust products.

Here's Thrust SSC going at Mach 1: enter image description here

You can see the shockwave going almost perpendicular to the vehicle, and the exhaust products (mixed with dust) that stay in a much narrower cone. More precisely you don't see the shockwave itself, but the dust that gets picked up off the desert floor by the shockwave.

As a final sanity check: if the Saturn V were moving at 1/3 its design speed at first stage separation, the mission would have failed because the CSM wouldn't have reached orbit. We know that didn't happen.

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    $\begingroup$ What's the big plume under the first stage, if not a contrail? The first-stage engines would be shut down by this point. $\endgroup$ – Russell Borogove Aug 14 '16 at 16:35
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    $\begingroup$ Without knowing the specifics of the Saturn V: shutdown wouldn't be instantaneous, the turbopumps wind down gradually. The stage would be separated as thrust drops under a set threshold. They might vent the remaining propellant through the pump and combustion chamber, too. $\endgroup$ – Hobbes Aug 14 '16 at 16:39
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    $\begingroup$ By the way, the Schlieren camera was not an invention of some (hypothetical) guy named Karl Schlieren (cousin of Fritz Eigen, inventor of eigenvalues). Instead, "Schlieren" is the German word for the optical effect and hence I suppose it should be a lowercased schlieren camera. $\endgroup$ – Hagen von Eitzen Aug 14 '16 at 17:37
  • $\begingroup$ Fair enough. Note that the exhaust plume of the second-stage engines constitutes a moisture-rich atmosphere; perhaps a visible shockwave can form in the plume but not further up at the other airframe diameter changes you mention. $\endgroup$ – Russell Borogove Aug 14 '16 at 18:15
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    $\begingroup$ So, in your own words, Apollo 11 moving at 1/3 of its design speed at stage separation would have had very obvious consequences. Those consequences didn't happen, so we can conclude Apollo 11 must not have been moving at 1/3 its design speed at stage separation. $\endgroup$ – Hobbes Dec 4 '16 at 20:43
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The main issue is that the image of the rocket is not taken from the side, but at an angle. We can determine the angle by looking at the apparent length to width ratio of the two stages - note that this screenshot has been taken directly after separation. The first stage seems to be twice as long as wide, the second stage is about three times longer than wide. Comparison with the actual size show a diameter of 10 m and a length of 40 m and 60 m respectively.

That means, all lengths seem to be shortened by a factor of 2. In particular, this makes the angle of the Mach cone seem to be twice as big as it actually is. After restoring the correct aspect ratio, I measure 18 degrees, corresponding to a Mach speed of 6.5, or 2200 m/s.

Given the poor quality of the image the number is reasonably similar to the expected 2700 m/s. As Hobbes pointed out, we can not be sure that it actually is a Mach cone, but is rather close to the expected one. I'm also in favor for the explanation as the outer edge of the second stage engine plume.

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    $\begingroup$ Kudos for the quantitative answer! More to read about "shock and mach" in The Wildly Misunderstood Aeronautics Event Captured in This Photograph and a very thorough, details discussion by NASA in Natural Visualization of Aircraft Flow Fields which has many photos. It is 150 pages - sometimes takes a while to load. $\endgroup$ – uhoh Aug 14 '16 at 3:44
  • $\begingroup$ I have added another picture, much more clear, with Apollo 11 just after the first stage separation.(see: upload.wikimedia.org/wikipedia/commons/e/e2/…). You can do all kind of precision measurements on this new photo. Tell me what you find. $\endgroup$ – Robert Werner Aug 15 '16 at 4:20
  • $\begingroup$ Using the new detailed picture I posted (see also my previous comment) and taking into account the corrections suggested by asdfex, the new angle I measured and also the decrease of the sound speed with altitude, I calculated a new speed at separation as ranging from 910 m/s to 1050 m/s. I do not get those 2200 m/s obtained by asdfex. $\endgroup$ – Robert Werner Aug 15 '16 at 22:37
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    $\begingroup$ The angle in the new picture seems slightly higher, i.e. 1800 m/s. I think this better picture shows us clearly that it is not a Mach cone we are looking at. Two things to keep an eye on: Don't include the bottom side of the rocket when measuring the length. Measure the angle in the straight part of the lines, not directly at the end of the rocket. There are too many dynamics involved in this area, and the shockwave can easily travel at supersonic speeds on the first couple of meters. $\endgroup$ – asdfex Aug 16 '16 at 8:08
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    $\begingroup$ @RobertWerner - nobody will prove what the speed was using these pictures. Both answers agree that this is very likely not a Mach cone! $\endgroup$ – asdfex Aug 17 '16 at 8:33
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At stage separation, the second-stage engine has begun its burn and an optimally-designed diverging cone of the nozzle will underexpand the jet of gas in the early part of the burn, causing wall separation within the nozzle where the shock wave forms. Consequently, the shock wave will be within the jet plume, and is usually visible as shock diamonds in the exhaust, the number of diamonds corresponding to the Mach number of the jet exhaust speed.

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  • $\begingroup$ Largely irrelevant to the question. $\endgroup$ – Organic Marble Jan 16 '17 at 1:43
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    $\begingroup$ The shock wave we're talking about is the one/ones caused by the rocket body, not by the exhaust. $\endgroup$ – Hobbes Jan 16 '17 at 8:51

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