This interesting, archived page https://www.webcitation.org/618QHms8h?url=http://www.fai.org/astronautics/100km.asp which I found in this answer, says:

In the early 1960´s, the U.S. X-15 Aircraft was flown up to 108 km. In that part of the flight it was really a free falling rocket, with no aerodynamic control possible. In fact, it was considered an astronautical flight, and the pilot got, as a consequence, his "astronautical wings", i.e. the recognition of being an astronaut.

A reading of this paragraph suggests that at or near this altitude, the X-15 had no attitude control available from aerodynamic surfaces nor propulsion (ergo the use of the expression "free-falling"). Was there in fact vector-able thrust available, or was it really just a hopefully-not-tumbling-very-much spacecraft near its periapsis? Or reaction wheels, or something else?

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    Strange: a quick skim of the Wiki page suggests that, at least near (and just above) the Karman line, there's insufficient lift to maintain altitude, but that doesn't strictly rule out roll/pitch/yaw control with the wing surfaces. Given the miniscule wings on the X-15, maybe there is just not enough area to have a significant effect on attitude (not altitude) – Carl Witthoft Oct 17 at 14:55
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    I suspect they switched to RCS below the Karman line. On jet fighters, aerodynamic controls become marginal at altitudes below 20 km. – Hobbes Oct 17 at 17:52
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    @Hobbes: they did switch to RCS after burn out at 42 km. But aerodynamic controls become marginal at altitudes ABOVE 20 km. – Uwe Oct 17 at 18:13
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    No, I meant below. "even at altitudes of less than 20 km, aerodynamic controls are already marginal. I can't imagine relying on them at 100 km" – Hobbes Oct 17 at 18:26
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    Without that context, it is definitely not clear English. – amI Oct 18 at 3:00

The X-15 had a reaction control system for all three axes using thrusters with hydrogen-peroxide monopropellant. There was an automatic as well as a manual mode. The manual mode used a single three-axis control joystick.

There were two completely independent systems. Each system used six RCS thrusters, two for each axis for both rotation directions. See this NASA paper. The image is from page 23.

enter image description here The RCS thrusters were used from a height of 140,000 feet or about 42 km up to 108 km. So the thrusters were used well below the Karman line. The transition from aerodynamic to reaction control was done immediately after main engine burnout.

There were problems with the corrosive nature of the 90 % concentrated H2O2. The only suitable materials for the tank bladder were teflon or a fluorosilicone rubber. The original RCS was constructed entirely of aluminium. After corrosion problems some parts were made of stainless steel. But then the use of two different metals resulted in electrolytic corrosion by the hydrogen peroxide. Eventually the entire system was made of stainless steel to eliminate the corrosion problem.

Hydrogen peroxide freezes at −0.43 °C (31.23 °F; 272.72 K). Because of the proximity of the H2O2 flow lines to the LOX tank of the X-15, wrap around heaters were used on all peroxide flow lines.

The X-15 has a reaction control system. In this image, it's item 2, 13 and 28, labeled 'ballistic control system'.

It was operated via a joystick.

X-15 general arrangement

Detail of two of the thrusters:

thruster detail

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