I have concluded that thermal transient effects are a likely explanation for the measured deflections and forces in NASA's torsion pendulum experiments of the Q drives. Explicitly, that they are the result of a shift in material location of the center of mass due to differential thermal expansion resulting from heating of the dielectric resonator which is positioned unsymmetrically. If this explanation is correct, Dr. White still should also be able to measure (slightly lower) forces when he places the Q drive in a torsion pendulum in a vacuum. However, if the Q drive were free in space (instead of supported from a pendulum), this transient, unsymmetric, thermal expansion would result only in a change in attitude (orientation).

I am posting here my letter to Dr. White in order to have a wider review of this explanation.

Dear Dr. White,

I have read with appreciable interest your paper (co-authored with D. Brady, P. March, J. Lawrence, and F. Davies) titled "Anomalous Thrust Production from an RF Test Device Measured on a Low-Thrust Torsion Pendulum", presented at the 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, July 28-30, 2014 in Cleveland, OH.

I have thought about what may be responsible for the measured displacement (and force) in your reported torsion pendulum experiments. Air convection resulting from microwave heating of the air surrounding the Q-drives has been suggested by various people, as the air speed that can produce the measured force can be shown to be small. You will be able to check whether air convection is responsible when you perform experiments in a vacuum (which I understand from your report was not possible because of the aluminum electrolytic capacitors that need to be replaced by capacitors that can work in a vacuum environment).

However, I have wondered how air convection could be responsible for the reproducible and fairly consistent levels of measured force pulse, as well as the fact that the experimental pulses are so well defined (and that it took practically no time to achieve the measured forces and to go back to zero upon ending the microwave pulse), and that turning the Q-drive around by 180 degrees resulted in practically the same force in the opposite direction.

Based on my experience conducting experiments at the Massachusetts Institute of Technology Aeronautics and Astronautics Department (for my S.B., S.M. and Ph.D. degrees at MIT) and later on at industrial R&D laboratories, I have arrived at the conclusion that transient thermal effects in your experiments should be carefully considered.

Indeed, after much thought and some calculations my conclusion is that the measured forces can quite likely be the result of transient thermal effects that very slightly shift the location of the center of mass in the material body of the Q-drive, due to unsymmetric thermal expansion, resulting from internal heating of the dielectric resonator in the Q-drive.

The center of mass changes location in the material body, with respect to body-fixed, Lagrangian coordinates, as it expands unsymmetrically. If the body would be free (unrestrained) in space, this would result only in a change in attitude (orientation) of the body. If free in space, the spatial position of the center of mass will not change (with respect to an inertial frame of reference). However, because the tested Q-drives were restrained, suspended from a support point in a torsion pendulum, the unsymmetric thermal expansion will generate a small measurable rotation and (torquing) force because the center of mass changes location in the material body as it expands unsymmetrically.

Similar issues (thermal distortion resulting in changes in orientation) were experienced, for example in spacecraft, most prominently with the Messenger (MErcury Surface, Space ENvironment, GEochemistry, and Ranging) spacecraft that got closer to the Sun: See:

http://messenger.jhuapl.edu/the_mission/publications/O'Shaughnessy_Pittelkau.2007.pdfInterestingly, your paper points out the importance of the (Teflon) dielectric resonator concerning the experimentally measured forces:

<<The longer beam pipe is the RF drive antenna that in practice ends up being a ¼ wave resonance system in its own right and has a dielectric PTFE slug in the throat in both the slotted and null test article. It is this characteristic that became an item of further consideration after completion of the test campaign.>> (p.7)

<<There appears to be a clear dependency between thrust magnitude and the presence of some sort of dielectric RF resonator in the thrust chamber.>> (p.18)

<<We performed some very early evaluations without the dielectric resonator (TE012 mode at 2168 MHz, with power levels up to ~30 watts) and measured no significant net thrust.>>( p.18)

It is noteworthy that the test conducted without the dielectric resonator resulted in no significant measurable force.

My analysis of the reaction force that results from the greater thermal expansion of the drive adjacent to the dielectric resonator shows that the reaction force should be in the direction towards the end that has no dielectric resonator. That is, if the dielectric resonator is to the right of the center of mass of the drive, the reaction force will be to the left, and if the drive is positioned such that the dielectric resonator is located to the left of the center of mass of the drive, the reaction force will be to the right. This agrees with all your experimental results.

My analysis of the reaction force that results from the greater thermal expansion of the drive adjacent to the dielectric resonator also shows that if the drive were perfectly symmetric (for example having the dielectric resonator centered at the center of mass, or having identical dielectric resonators located at the same distance from the center in both directions), there would be no net thermal distortion forces as they would balance themselves out. In other words, if the Q-drive would have the dielectric resonator in the middle, or have two identical dielectric resonators positioned at equal distances from the center of mass, there would be no measurable forces.

Heat is generated inside the dielectric resonator due to the dielectric loss ("tan delta") material property of the resonator. This internal heat power is produced instantly as a result of the electromagnetic field but it takes a finite amount of time for the temperature to diffuse through the material and reach steady state in accordance with Fourier's equation of heat conduction, depending on the diffusivity of the material, and satisfying the thermal boundary conditions (convection and radiation if the experiment takes place in air, and just radiation if it takes place in a vacuum). Since the dielectric loss factor ("tan delta") is temperature dependent, the heat generated is also temperature-dependent, which introduces a nonlinearity in the solution of the differential equations for this problem. As the polymer ("Teflon" PTFE thermoplastic Fluoropolymer) dielectric temperature rises, it expands both in its radial and longitudinal direction. There is also a dynamic effect due to the inertial forces reacting to the sudden pulse, in addition to the torsional resisting force of the torsional pendulum.

The thermal expansion of the polymer dielectric resonator in the radial direction results in better contact and heat transmission to the copper structure of the drive. It can be readily shown that the effect of air convection in this experiment should be small in comparison with thermal conduction.

The reaction force produced by unsymmetric thermal expansion is proportional to the second derivative of temperature with respect to time.

To calculate how long it takes for the temperature distribution to reach steady state (and therefore for the second derivative of temperature with respect to time to become negligibly small) we may use the Fourier Number: the thermal diffusivity times the characteristic time divided by the square of the characteristic length. It is known that steady state is typically reached for a Fourier number exceeding unity, that is, for the characteristic time exceeding the ratio of the square of the characteristic length divided by the thermal diffusivity. The thermal diffusivity of Teflon is 0.124 (mm^2)/sec. I could not find the dimensions of the Teflon dielectric resonator in the report. I calculate that the time to reach thermal steady state exceeds 22 minutes if the characteristic length of Teflon is 0.5 inch (12.7 mm). If the characteristic length of Teflon is 0.2 inch (5 mm), the time to reach steady state will exceed approximately 4 minutes. If the characteristic length of Teflon is 1 inch (25.4 mm), the time to reach steady state will exceed 1 hour and 27 minutes. We know from the report that the microwave pulse was maintained for only 35 seconds during the testing (see Fig.12, p.9 in the report). Therefore, we know that the microwave pulse was maintained for an amount of time much shorter than the amount of time necessary for the temperature distribution to reach steady state in the Teflon dielectric resonator.

When the microwave power is turned off (Fig.12, p.9 of the report shows this happening 35 sec after it was turned on), the heat generating power suddenly becomes zero, and hence the second derivative of the temperature with respect to time (responsible for the reaction force) becomes negative when the microwave power is turned off, resulting in a force in the opposite direction as to when the microwave power was on.

Since copper's Young modulus is about 300 times stiffer than Teflon's, and assuming that the Teflon, particularly as it expands radially, is in frictional contact with the surrounding copper, it makes sense to assume that the expansion of the Teflon dielectric resonator is restrained by the much stiffer copper. Under that assumption, we can calculate the differential thermal expansion of the copper surrounding the Teflon as the product of the coefficient of thermal expansion of copper (16.6*10^(−6) 1/degC) times the longitudinal length of the Teflon resonator, times the "delta T": the temperature difference between the copper surrounding the Teflon and the rest of the structure.

If the longitudinal length of the Teflon resonator is 1 inch (25.4 mm), the delta T necessary to produce a differential thermal expansion of 4 micrometers is only 9.5 deg C (17 deg F). So it is quite possible to produce the measured deflections with a delta T in temperature of a few deg C. If the Teflon is unrestrained by the copper, the required delta T is 8 times smaller (since the coefficient of thermal expansion of Teflon is 135* 10^(−6) 1/degC, eight times greater than the coefficient of thermal expansion of copper).

I hope that these considerations, convince you (as has been my experience in testing at MIT and in industrial R&D) that thermal transient effects are important and therefore that it merits strong consideration that the measured deflections and forces in your torsion pendulum experiments of the Q drives are the result of a shift in material location of the center of mass due to differential thermal expansion resulting from heating of the dielectric resonator which is positioned unsymmetrically, as explained above.

Best regards,

Dr. Jose' J. Rodal

jrodal@alum.mit.edu