I think this is a contrail effect just like those caused by aircrafts.
Both aircraft jet fuel and rocket fuels like RP-1 and liquid hydrogen produce a lot of water vapor when burned within the respective engine.
Depending on the temperature and humidity at the altitude the
contrails form, they may be visible for only a few seconds or minutes,
or may persist for hours and spread to be several miles wide,
eventually resembling natural cirrus or altocumulus clouds.
At high altitudes as this water vapor emerges into a cold environment,
the localized increase in water vapor can raise the relative humidity
of the air past saturation point. The vapor then condenses into tiny
water droplets which freeze if the temperature is low enough. These
millions of tiny water droplets and/or ice crystals form the
contrails. The time taken for the vapor to cool enough to condense
accounts for the contrail forming some distance behind the aircraft.
At high altitudes, supercooled water vapor requires a trigger to
encourage deposition or condensation.
Cites from wikipedia.
The rocket engine nozzle expansion does not influence the production of water vapor.
NASA photo of a X-15 contrail from this page.
From this page.
This layering of the air into wet and dry layers is not limited to
clouds. Seemingly clear air also contains exactly the same kind of
variation in layers. This was very neatly illustrated by the recent
launch of the Solar Dynamics Observatory. As it ascended it did not
leave a contrail, until it hit a layer of wet air, when it left a
contrail that lasted quite a while, and then it went into dry air
again, and no more contrail.
From this page.