NASA Brief: Q-Thruster Physics June 2013 Dr. Harold "Sonny" White • Q-thrusters are a low-TRL form of electric propulsion that operates on the principle of pushing off of the quantum vacuum. • A terrestrial analog to this is to consider how a submarine uses its propeller to push a column of water in one direction, while the sub recoils in the other to conserve momentum – the submarine does not carry a "tank" of sea water to be used as propellant. • In our case, we use the tools of Magnetohydrodynamics (MHD) to show how the thruster pushes off of the quantum vacuum which can be thought of as a sea of virtual particles - principally electrons and positrons that pop into and out of existence, and where fields are stronger, there are more virtual particles. • The idea of pushing off the quantum vacuum has been in the technical literature for a few decades, but to date, the obstacle has been the magnitude of the predicted thrust which has been derived analytically to be very small, and therefore not likely to be useful for human spaceflight. • Our recent theoretical model development and test data suggests that we can greatly increase the magnitude of the negative pressure of the quantum vacuum and generate a specific force such that technology based on this approach can be competitive for in-space propulsion (~0.1N/kW), and possibly for terrestrial applications (~10N/kW). • As an additional validation of the approach, the theory allows calculation of physics constants from first principles: Gravitational constant, Planck constant, Bohr radius, dark energy fraction, electron mass. • Local mass concentrations, say in the form of a conventional capacitor with a ceramic dielectric, affect vacuum fluctuation density according to equation 1: ρ_v_local = ρ_v * sqrt(ρ_m_local / ρ_v) = sqrt(ρ_m_local * ρ_v) (1) • Just as relativistic acceleration (Unruh radiation) can change the apparent relative density of the vacuum, so too can higher order derivatives according to equation 2: δρ = (1 / (4πG)) * [ - (1/a^2)(da/dt)^2 + (1/a)(d^2a/dt^2) ] δρ = (1 / (4πG)) * [ (1/ϕ^2)(dϕ/dt)^2 - (1/ϕ)(d^2ϕ/dt^2) ] (2) where a = -∇ϕ • The tools of MagnetoHydroDynamics (MHD) can be used to model this modified vacuum fluctuation density analogous to how conventional forms of electric propulsion model propellant behavior. • Consider the following thought experiment: what would an inertial observer in deep space find if the dark energy density were to be integrated over the light horizon of the observable universe, ~13.7 billion light years? • Starting with the Friedman Equation (and after some manipulation), the following equation can be derived that formally captures the results of this thought experiment: (2/3) * ρ_0 * c^2 * 4π * c^2 * t_H^2 = c^4 / G • Equation can be rearranged into the following form: G = 1 / (4π * t_H^2 * (2/3) * ρ_0) • Using 9.9x10^-27 kg/m^3 with t_H of 13.7 billion years yields a predicted value for the gravitational constant of 6.45x10^-11 m^3/kg·s^2. • A possible physical meaning to this rearranged equation solved for G is that gravitation is an emergent phenomenon rather than a fundamental force. • To be specific, the claim could be made that the gravitational coupling constant may be a long wavelength consequence (λ=ct_H) of dark energy. • Start with the Friedmann Equation, and through some manipulation, the gravitational coupling constant, G, can be shown to be a time-independent function of the dark energy density integrated over the spherical light horizon. Recall the Friedmann Equation (1): [(1/R)(dR/dt)]^2 - (8πGρ_0 / (3R^3)) * R^2 = -k c^2 (1) • The variables take on their familiar nomenclature: R is the co-moving coordinate; t is time; G is the gravitational constant; ρ_0 is the critical density; k is the curvature; and c is the speed of light. Assume a flat universe, k=0, and simplify to get (2): (dR/dt)^2 - (8πGρ_0 / 3R) = 0 (2) • This can be solved for R to get the following familiar time-dependent function (t_H is the Hubble time, or rather the age of the universe) (3): R = (3/2)^(2/3) * (t / t_H)^(2/3) (3) • Going back to equation 2, and set up the integrals again (4): ∫ R^(1/2) dR = ∫ sqrt(8πGρ_0 / 3) dt (4) • This can be integrated to yield (5): (2/3) R^(3/2) = t * sqrt(8πGρ_0 / 3) (5) • Simplify and rearrange to get (6): (4/9) R^3 = (8πGρ_0 / 3) * t^2 (6) • G can be isolated to one side (7): (9/4) * (2/3) * (4πρ_0 * t^2 / R^3) = 1/G (7) • Equation 3 can be used to substitute for the co-moving coordinate, R, to get (8): (9/4) * (2/3) * 4πρ_0 * t^2 * (2/3)^2 * (t_H / t)^2 = 1/G (8) • Cancelling terms and multiply both sides by c^4 (9): 4π c^2 t_H^2 * (2/3) ρ_0 c^2 = c^4 / G (9) • Second, although equation 9 has the Hubble time (current age of the universe) in it, t_H, it is not a function dependent on time and is constant for the duration of the cosmos as the variable time dependency cancelled in equation 8: G = [ 4π * t_H^2 * (2/3) ρ_0 ]^-1 (9b) • Consider the Einstein Tensor: The Gravitational Coupling Constant equation derived earlier can be substituted for the Planck Force term in the 00th element of the Einstein Tensor, and the dark energy density can be substituted for the T^00 energy density term. Plugging in the Hubble time, the following relationship is found to be numerically true: ⟨0| G^μν |0⟩ = -2 / (c^2 t_H^2) = -h R / (4π t_H) • This equation can be rearranged to solve for the Planck constant, h, which yields a predicted value of 6.46x10^-34: h = 8π / (c^2 t_H) • This equation suggests that the basic geometry of the cosmos is related to the lowest energy observable photon with same wavelength discussed in the derivation of the gravitational coupling constant. • These tiny building blocks or "lego" bits of energy may be a phenomenological explanation for the nature of the Heisenberg uncertainty principle. • The previous narrative has been focused on things at a cosmological scale. A question can be asked about what happens on a local scale in the presence of a local concentration of matter. • How does dark energy, or more appropriately the quantum vacuum, respond in the presence of ordinary matter? • Start by reconsidering the Einstein tensor in the presence of a local concentration of ordinary matter. The 1/t_H term on the right side of the Planck equation is replaced by the local conditions imposed by the presence of local matter with a characteristic spacing defined as Δx. Note that the Gravitational Coupling Constant equation has been used to substitute for the Planck force c^4/G: G^00 = -8π (G/c^4) T^00 = -8π [ 1 / ((2/3) ρ_0 c^2 * 4π c^2 t_H^2) ] ρ_v_local c^2 = hc / (4π Δx) • Equation 1 can be used to redefine the characteristic spacing Δx in terms of the local matter density with the assumption that gravitation is constant at all scales. • The long wavelength c t_H term is replaced by the maximum wavelength allowed Δx in the presence of local matter density ρ_m: ρ_m c^2 4π Δx^2 = c^4 / G ==> Δx^2 = c^2 / (4π G ρ_m) • These two equations can be used together to derive the perturbation of the local quantum vacuum state as a result of the local matter concentration. The ρ_v term has been substituted for the 2/3 ρ_0 term for brevity: ρ_v_local = sqrt(ρ_m * ρ_v) • A possible physical interpretation of the local perturbation of the quantum vacuum in the presence of ordinary matter is provided for consideration. • In the Casimir cavity scenario, the presence of two conducting plates in very close proximity perturbs the local quantum vacuum state between the plates such that the pressure between the two plates is negative relative to the pressure outside the plates resulting in a force that pushes the plates together. • As the separation distance between the two plates 'a' becomes comparable to the inter-atomic spacing in the lattice structure of the plates 'd', the two plates begin to approximate a single monolithic plate rather than two distinct plates. • In the limit of this scenario, the Casimir force does not know when to "shut off", and the perturbed vacuum state will still have a negative pressure when compared to the unperturbed state. • Similarly within the atomic lattice structure of either plate, there are infinite combinations of virtual Casimir cavities that can be traced out owing to the crystalline structure of the atomic nuclei. • Said simply using the visualization tools of nuclear engineering, if quasi-classical pressure readings could be made from a miniature "atomic person" peering between the two plates with a quantum vacuum "pressure gauge", the "atomic person" would find a similar perturbed negative pressure state of the quantum vacuum within the lattice structure of either plate. • The quantum vacuum responds due to the presence of the two plates creating a distinct Casimir cavity, and similarly the quantum vacuum responds due to the presence of the virtual Casimir cavities defined by the crystalline structure of the local concentration of matter. • The vacuum perturbation equation just derived can be used to evaluate the state of the quantum vacuum in close proximity of the proton at the center of the Hydrogen atom. • The first step is to calculate a quasi-classical density for the hydrogen nucleus. The radius of the hydrogen atom nucleus is given as R_0 = 1.2x10^-15 m (R = R_0 · A^(1/3) where R_0 = 1.2x10^-15 m and A is the atomic number - these are experimentally determined by electron scattering). • The radius can be used with the mass of a proton to calculate a quasi-classical density of the hydrogen nucleus: ρ_m = m_p / ((4/3) π R_0^3) = 2.31x10^17 kg/m^3 • Using ρ_v = (2/3) * 9.9x10^-27 kg/m^3, along with this quasi-classical density ρ_m, the perturbed negative pressure state of the quantum vacuum around the hydrogen nucleus is calculated to be: ρ_v_local = sqrt(ρ_m * ρ_v) = 3.90x10^-5 kg/m^3 • The question can be asked how much volume of this perturbed state of the quantum vacuum is needed to have the equivalent energy value as the ground state of Hydrogen (13.6eV or 2.18x10^-18 Nm): r = [ E / (ρ_v_local c^2 * (4/3) π) ]^(1/3) = a_0 • The calculated radius is r = 5.29x10^-11 m, which is an exact match to the given value for the Bohr Radius, a_0 = 5.29x10^-11 m. • The derivation of the gravitational coupling constant included a macroscopic integration of the dark energy density over the light horizon. What happens if an analogous microscopic exercise is performed for the vacuum density present around the hydrogen atom? If the 2/3 factor is included, the result is the electrostatic force: 4π a_0^2 * (2/3) ρ_v_local c^2 = q^2 / (4π ε_0 a_0^2) • Similarly, the energy density of the electric field around the atom can be compared to the energy density of the vacuum density. Note that here the equation is exact only if one-third of the vacuum density is used: (1/3) ρ_v_local c^2 = E ε_0^2 / 2 • Both of these equations suggest that some of the tools of MHD can be used to model the behavior of the vacuum around the hydrogen atom, suggesting that some focus also be put on the magnetic field to see if this hypothesis remains consistent. • Frank Wilczek, Nobel laureate: "We have achieved a beautiful and profound understanding of the origin of most of the mass of ordinary matter, but not of all of it. The value of the electron mass, in particular, remains deeply mysterious…" • Consider the energy state of the perturbed quantum vacuum field around the proton, and set this equal to the kinetic energy of the orbiting electron at the ground state: (4/3) π a_0^3 ρ_v_local c^2 = (1/2) m_e v^2 • We know the speed of the orbiting electron: v = α c = c / 137 • We can solve for the electron mass, and using the predicted value for ρ_v_local of 3.9x10^-5 kg/m^3, we get a predicted electron mass of 9.1x10^-31 kg: m_e = ( (8/3) π a_0^3 ρ_v_local c^2 ) / (c / 137)^2 • Local mass concentrations, say in the form of a conventional capacitor with a ceramic dielectric, affect vacuum fluctuation density according to equation 1: ρ_v_local = sqrt(ρ_m_local * ρ_v). • Just as relativistic acceleration (Unruh radiation) can change the apparent relative density of the vacuum, so too can higher order derivatives according to equation 2. • The tools of MagnetoHydroDynamics (MHD) can be used to model this modified vacuum fluctuation density analogous to how conventional forms of electric propulsion model propellant behavior. • Pioneer Venus Orbiter (PVO) routinely traversed regions of extreme low plasma pressure (holes) on the night side of Venus enabled by the presence of enhanced magnetic pressure: P = n_e k T <==> B^2 / (2 μ_0) = n_e k T • The first step now is to calculate the magnetic pressure around the Hydrogen nucleus. • The magnetic field as perceived by the electron is given by B = μ_0 q v / (4π a_0^2), where the speed of the orbiting electron is αc. • The magnetic pressure is a simple calculation: B^2 / (2 μ_0) = 6.25x10^7 N/m^2. • The quasi-classical plasma pressure of the perturbed quantum vacuum state around the Hydrogen nucleus can be calculated by converting the electron velocity to temperature using (1/2) m_e v^2 = kT, and making the assumption that the virtual electron-positron plasma has the same effective temperature as the orbiting electron. • When the plasma pressure calculation makes use of a 2/3 factor, analogous to the predicted dark energy fraction of 2/3 picked up during integration to calculate the Gravitational constant, the values are nearly identical: P = n_e k T = (2/3) * (ρ_v_local / m_e) * k T = 6.24x10^7 N/m^2 • Consider the extragalactic magnetic field which is estimated to be 1x10^-12 Tesla. If the quantum vacuum can be treated as a virtual plasma, then the magnetic energy density (or pressure) should correlate to the plasma pressure. The magnetic pressure is calculated using the following equation: P_B = B^2 / (2 μ_0), B = 1x10^-12 T, μ_0 = 1.26x10^-6 T^2m^3/J, P_B = 3.98x10^-19 N/m^2. The plasma pressure can be calculated using the following equation: P_plasma = n_e k T. The electron-positron density n_e can be found using n_e = ρ_c / m_e. The critical density is as stated before, ρ_c = 9.9x10^-27 kg/m^3, and the temperature is T = 2.73K. Assuming an electron-positron plasma population, the plasma pressure is P_plasma = 4.09x10^-19 N/m^2. • This relationship suggests that in the far field limit, the magnetic field affects the quantum vacuum. This same methodology can be applied to dark matter models for galaxies to see if there is a similar correlation when treating dark matter as a virtual electron-positron plasma. Current dark matter models for galaxies can be used to predict a galactic halo magnetic field as a function of galactic radius, and this magnetic field magnitude distribution can be compared to observation. Although galactic halo magnetic field strength and structure is not fully understood, the predictions can still be compared to the data and models available [8]. Figure 4 shows the comparison. As with the extragalactic magnetic field, there is a very strong correlation for the galactic magnetic field. • These findings also show that the tools of MHD can successfully be used to model quasi-classical behavior of the vacuum. • Local mass concentrations affect vacuum fluctuation density: ρ_v_local = sqrt(ρ_m_local * ρ_v). • Unruh acceleration and potential dynamics: δρ = (1 / (4πG)) * [ - (1/a^2)(da/dt)^2 + (1/a)(d^2a/dt^2) ]. • MHD modeling of modified vacuum fluctuation density and virtual plasma drift under orthogonal E and B fields. • The dynamic Casimir force arises as a result of Unruh radiation where an accelerated observer sees the vacuum as a higher temperature photon bath, and is the mechanism that facilitates Hawking radiation around a black hole where relativistic acceleration separates a virtual pair such that one particle goes in the horizon, while the other escapes. • Recent findings reported earlier in 2011 show that the dynamic Casimir effect may have been detected in the lab: http://www.technologyreview.com/blog/arxiv/26813/ • The simplest mechanical construct to help visualize using the dynamic Casimir force to generate thrust is through the use of vibrating mirrors where the mirror trajectory is designed to generate radiation in a preferred direction. • The magnitude of thrust arising from using the dynamic Casimir force derived numerous times in the literature has been shown to be very small in comparison with conventional propulsion systems, but has been clearly shown to be theoretically possible. • The discussion will now continue on to derive a wave equation that relates perturbation of the quantum vacuum to first and second order time-varying potential terms. • Start with the Friedmann equation (A): (1/R^2)(dR/dt)^2 = 8πGρ / 3 • ...then using the fact that RT = constant [1] and incorporating Birkhoff's theorem [1] (B): (1/R)(dR/dt) = - (1/T)(dT/dt) • Take the second derivative and simplify (C): (1/R)(d^2R/dt^2) = - (1/T) [ - (2/T)(dT/dt)^2 + (d^2T/dt^2) ] • Birkhoff's theorem can be restated to facilitate substitution (D): (1/R)(d^2R/dt^2) = - (4/3) πGρ • D can be subtracted from A to yield (E): (1/R^2)(dR/dt)^2 - (1/R)(d^2R/dt^2) = (8/3) πGρ + (4/3) πGρ • B and C can be substituted into E to recast the discussion from R to T (F): - (1/T^2)(dT/dt)^2 + (1/T)(d^2T/dt^2) = 4πGρ • At this point, let us consider Unruh radiation - Unruh radiation is conjectured thermal radiation emitted by the vacuum as detected by an accelerated observer. • Said differently, an observer accelerated with respect to the vacuum will see themselves immersed in a thermal bath - the higher their acceleration, the higher the temperature of the thermal radiation. This relationship is defined as: kT = ħ a / (2π c) ==> T = ħ a / (2π c k) (G) • In response to the proper acceleration a, the accelerated observer will see the vacuum take on the temperature, T. • Taking the first and second time derivatives of G, and substituting these relationships back into F: δρ = (1 / (4πG)) * [ - (1/a^2)(da/dt)^2 + (1/a)(d^2a/dt^2) ] δρ = (1 / (4πG)) * [ (1/ϕ^2)(dϕ/dt)^2 - (1/ϕ)(d^2ϕ/dt^2) ] where a = -∇ϕ • This equation relates perturbation of the quantum vacuum to first and second order time-varying potential terms. • Baseline geometry of matter has average lattice spacing. • Addition of photon number density due to application of high power RF radiation decreases average spacing between "boundaries" in quantum vacuum. • This gives the matter the appearance of having more "molecules" per given volume which simulates a higher density, and increases the perturbation of the vacuum to a more negative state. Summary of Test Articles and Findings: 1. 2004 Test Article: ~4 mN Thrust, Specific Force ~0.4 N/kW. 2. 2005 Test Campaign: Run at 2.13 MHz (20 kV/m, 27 Gauss) with observed thrust +/- 0.89 mN (prediction 0.63 mN); and 3.8 MHz (20 kV/m, 48 Gauss) with observed thrust +4.91 to -1.96 mN. Specific Force ~0.3 N/kW (~3 mN thrust). 3. 2012 Test Article: Tested November 2012; 98 μN predicted, 2-3 μN detected; vacuum fluctuation density increased from ~1x10^-26 to >1x10^-14. 4. SFE Test Article at JSC (2013): Evaluated with Boeing/DARPA in/out of Faraday Shield (Feb-June 2013). Consistent transient thrust pulses ~20-110 μN scaling with cube of input voltage, specific force ~1-20 N/kW. 5. Microwave Thruster Device (SPR Ltd / Shawyer): 16-170 mN thrusters, specific force 0.02-0.4 N/kW (Prototype: 16mN @ 850W; Dynamic: 96mN @ 334W; High fidelity: 170mN @ 450W). Tapered shape creates virtual toroid of active volume. 6. Cannae Test Article: Superconductive Niobium-Tin resonant cavity, ~7-10 mN thruster, specific force ~0.75 N/kW. What we suspect: • A variety of industry experiments, for which theory is lacking, may be Q-thrusters including Boeing, Lockheed-Martin, EM Drive, Cannae, etc. • Low measured thrust but specific power ranges from 0.3 to 10+ N/kW. Propulsion Technology Roadmap across application domains: • In-space (~0.1 N/kW): ISS Demo (CMG perturbation), ISS Free Flyer, COTS Free Flyer, Class D Mission. • Aero / High-altitude (~10 N/kW): High altitude, high speed aircraft; LOX/LCH4 fed turbines powering banks of q-thrusters; Blimp Test. • Space-lift / Orbit transfer: High thrust, high Isp spacelift (2000s effective specific impulse); Orbit transfer; Space Tugs. • Outer Solar System & Deep Space: Fast Mars missions, Outer solar system exploration & beyond, Repetitive powered overflight of ISR targets (akin to Space UAV). Analysis of transit times to outer planets (Jupiter, Saturn, Uranus, Neptune, Pluto) using Quantum Vacuum Plasma Thrusters (QVPT) at 0.4 N/kW: • Performance based on tuned 2005 test article performance: 4000 microN / 10W input (0.4 N/kW). • Specific mass of power system: 10 kg/kW; Specific mass of propulsion system: 10 kg/kW. • Configurations evaluated: P = 2 MW (50 mT payload), P = 5 MW (50 mT payload), P = 10 MW (50 mT payload). • Mission profiles assume positive capture at target (accelerate first half of voyage, decelerate second half; not a flyby). • Transit times range from ~100-200 days to Jupiter/Saturn to ~300-600 days for Neptune/Pluto. Backups section title slide. • Recent models developed by White suggest that there are ways to increase the net force, and these models have been validated against data at both the cosmological scale, the quantum level, and test devices: – Claim 1: An increase in local matter density correlates to an increase in local quantum vacuum energy density according to ρ_v_loc = (ρ_v * ρ_m)^0.5. (Substantiated by exact prediction of Bohr radius). – Claim 2: The energy density of the quantum vacuum can be amplified by acceleration and higher order time derivatives. Derived using FRW metric with Unruh radiation equation, aligning with Sciama (ϕ=c^2) and Woodward. – Claim 3: The quantum vacuum can be modeled as electron-positron virtual plasma (Dirac vacuum), allowing MHD tools to predict macroscopic behavior. (Substantiated by galactic magnetic fields, dark matter correlations, hydrogen atom plasma pressure = magnetic energy density, and vacuum permittivity/permeability ε_0 and μ_0 origin). Continuity Equations across virtual plasma shock at the edge of capacitor: • Conservation of Mass: ρ_1 * v_1 = ρ_2 * v_2 • Momentum Conservation: ρ_1 * v_1^2 + P_1 + (B_1^2 / (2 μ_0)) = ρ_2 * v_2^2 + P_2 + (B_2^2 / (2 μ_0)) • Energy Conservation: v_1 * [ (1/2) ρ_1 v_1^2 + (γ P_1 / (γ - 1)) + (B_1^2 / μ_0) ] = v_2 * [ (1/2) ρ_2 v_2^2 + (γ P_2 / (γ - 1)) + (B_2^2 / μ_0) ] • Possible mechanisms to satisfy conservation of mass: false vacuum, superluminal phonons, higher dimensional space-time geodesics (Chung & Freese, 2000; Stevenson, 2004). NASA Eagleworks Laboratories Poster Summary: • Mission Statement: Human exploration and colonization of the Solar System in the next 50 years and robotic interstellar flight by the end of the 21st century. • Quantum Vacuum Theory: ZPF ground state energy integration, COBE sphere vacuum energy density, Casimir force predictions. • Principles of Q-Thruster Operation: Using MHD equations of motion with virtual plasma drift (E x B drift); high-density dielectric and high-frequency electric field boosting vacuum density from 1x10^-27 kg/m^3 to 1x10^-12 kg/m^3. • Target Goals: Scalable thrust from milli-Newtons to Newtons for ISS RCS testing (0.1-1 N at 0.3-3 kW); enabling fast missions to Mars (30-37 days), Jupiter (112-135 days), and Proxima Centauri (10 MW system arriving in ~100 years).