Below are some screen shots from circa June 2016 launch (or re-launch) videos by Blue Origin (video link) and SpaceX (video link). The New Shepard burns $\text{H}_2$/$\text{O}_2$, while the Falcon 9 burns Kerosene/$\text{O}_2$.

I believe that at least part of the intense white/yellow light from the Kerosene/$\text{O}_2$ exhaust comes from thermal emission of Carbon particulates (soot) similar to the emission from the white/yellow part of a candle flame. There is very little luminescence from the $\text{H}_2$/$\text{O}_2$ exhaust.

The first image in the question Why do the exhaust flames from cryogenic stage engines appear to be separated from the nozzle? is quite striking, showing a very sudden appearance of brilliant light when the exhaust is compressed by a static shock wave. Presumably this is $\text{H}_2$/$\text{O}_2$ but I couldn't see a link or citation for the image.

Left (below) While candle flames also emit blue light - this is said to be mostly from Swan band emission - vibrational transitions in molecular Carbon as $\text{C}_2$. These do not occur in $\text{H}_2$/$\text{O}_2$ combustion. You can see Swan band emission stimulated by solar radiation in molecular carbon from comets as well - here is comet C/2014 Q2 (Lovejoy) (larger size available here).

Right (below) Image of Space Shuttle Main Engine test firing from here (larger size available here). Note the sudden appearance of bright light as the exhaust reaches the shock wave near the bottom of the image.

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The answer there that I found the most helpful discusses the shock wave and the blue emission from $\text{H}_2$/$\text{O}_2$ exhaust. So I went back to the New Shepard launch video and sure enough - you can see what looks like blue light from the nozzle when viewed from below.

What actually produces this blue light? Is it $\text{H}_2 \text{O}$ emission, $\text{OH}^-$ emission, or something else? And why does it only appear at higher densities?

side note: see $\text{O}_2$ flame burning in $\text{H}_2$ and mixture ratio optimization.

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As pointed out by @Thomas, this nice thesis Radiation from High Pressure Hydrogen-Oxygen Flames and its Use in Assessing Rocket Combustion Instability - Ph. D. Thesis, Fiala, T., 2015 discusses this phenomenon. The term $\color{blue}{\text{blue radiation}}$ is suggested as the best available.

While there are several narrow spectral lines int the near UV (about 310nm) due to excimeric hydroxyl radical $\text{OH}^*$, the blue light in the visible part of the spectrum appears to be a broad, unresolved continuum.

Figure 5.2 from Fiala 2015 shows low resolution spectra from $\text{H}_2 /\text{O}_2$ flames at different pressures. The nonlinearity of the intensity vs pressure can be seen in the substantial jump in intensity around 450nm as the pressure is increased from 21 to 30 bars ('atmospheres'). This may be the same onset that produces the sudden appearance of the blue radiation below the engine in the test-firing images in the question, where a first shock-wave produces a volume of sudden high pressure.

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In Section 5.3. Investigation of the Origin of the Blue Radiation the two most like sources of blue chemiluminescence are presented and discussed:

Blue Radiation

Through spatial imaging of the luminescence at different pressures, combined with numerical modeling of the kinetics and evolution of the processes within the flame, Fiala 2015 concludes that only the second reaction (5.2) leading to the production of chemiluminescence from excimeric $\text{H}_2\text{O}_2^*$ is consistent with all of the current experimental results.

The Space Shuttle provides a nice contrasting view of both the intense yellow light presumably from some combination of blackbody radiation soot and the aluminum reaction products in the SRB's exhaust, and the blue radiation from the water-like exhaust from the shuttle's main LH2/LOX engines. The pronounced first static shock wave followed by shock-diamond-like repeats below are clearly evident.

enter image description here above: STS-123 (NASA) from here - cropped. The vertical white bar below the shuttle is possibly an anti-lightning mast, and not an exhaust plume. Guy wires can be seen as well.

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    $\begingroup$ I thought the principal fuel in the SRBs is aluminum, creating a lot of aluminum oxide particulate, the bright white light was coming from that. Not correct? $\endgroup$ – Anthony X Mar 24 '18 at 13:45
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    $\begingroup$ I understand the focus of the OP's question relates to the blue light from the SSMEs, but in your answer you stated the contrast with the light emitted by the SRB exhaust, saying it is carbon-based, which I don't believe to be entirely true. If you compare the exhaust plume from the SRBs with that of an RP-1/LOX booster, say Falcon 9 or Saturn V, the carbon-based fuels produce a noticeably yellow plume, where the SRB plume is a bright white, which I understood to be from the (primarily) aluminum fuel. $\endgroup$ – Anthony X Mar 24 '18 at 16:57
  • $\begingroup$ @AnthonyX Oh! This is almost two years old, and I missed the fact that I'd answered my own question, (thought this was one of my early, extremely long questions), sorry about that! Okay I'll adjust the wording for now and see what I can do to find out more about the color. Thank you for bringing this to my attention! I thought there might be something I could link to at the question How do rocket propellant combinations rank in terms of “brightness”? but I don't see anything conclusive there yet. $\endgroup$ – uhoh Mar 25 '18 at 1:05
  • $\begingroup$ @AnthonyX I've just asked What makes exhaust from aluminum-based SRB propellant so bright? $\endgroup$ – uhoh Mar 25 '18 at 4:08

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