Thrust-to-weight ratio
The thrust-to-weight ratio (TWR), also denoted as T/W or F/W, is a dimensionless engineering parameter that quantifies the thrust generated by a propulsion system relative to the weight of the vehicle, equating to the acceleration capability in multiples of Earth's gravitational acceleration (g ≈ 9.8 m/s²).[1] This ratio is calculated as F/W = a/g, where F is thrust, W is weight (mass times g), and a is acceleration, directly linking propulsion efficiency to vehicle performance per Newton's second law.[1] In aerospace applications, a TWR greater than 1 enables vertical ascent or rapid climb, as excess thrust overcomes gravity and drag, while values below 1 are typical for efficient cruise in commercial aircraft.[1] For aircraft, TWR is a critical design factor in constraint analysis, influencing takeoff, climb rate, and maneuverability; for instance, fighter jets achieve TWR near 1.0 for superior agility, whereas airliners operate around 0.3 for fuel efficiency during level flight where T/W ≈ 1/(L/D), with L/D being the lift-to-drag ratio.[2] In rocketry, TWR must exceed 1 at liftoff to achieve ascent, typically around 1.2 to 1.5 initially for launch vehicles, rising as propellant mass decreases, and it expresses the engine's ability to accelerate the payload and structure.[3] Measured under standard conditions like sea-level static thrust, TWR varies with altitude and fuel load, guiding trade-offs between power, weight, and mission requirements in propulsion system design.[1]Fundamentals
Definition
Thrust is the mechanical force generated by an engine or motor that propels a vehicle forward, counteracting drag and enabling motion through the air or space.[4] In the context of aerospace vehicles, weight represents the gravitational force exerted on the vehicle's mass, calculated as W = m g, where m is the mass of the vehicle and g is the acceleration due to gravity, approximately 9.81 m/s² at Earth's surface.[5] The thrust-to-weight ratio (TWR) is a dimensionless parameter defined as the quotient of the thrust produced by the propulsion system divided by the vehicle's weight, quantifying the extent to which the thrust surpasses the gravitational pull on the vehicle—essentially, how many times greater the propulsive force is compared to the weight.[1] A TWR greater than 1 allows vertical ascent against gravity, while values below 1 indicate insufficient thrust for liftoff under static conditions.[1] The term gained formal prominence in engineering literature during the mid-20th century, as exemplified in George P. Sutton's influential textbook Rocket Propulsion Elements, which standardized its use for evaluating propulsion performance.[6] Unlike the power-to-weight ratio, which assesses the engine's power output (in watts or horsepower) per unit mass and is particularly relevant for propeller-driven systems where energy delivery influences sustained speed, the thrust-to-weight ratio specifically emphasizes the instantaneous force provided by reaction engines relative to gravitational load, making it critical for acceleration and vertical flight capabilities in jets and rockets.[1][7]Calculation
The thrust-to-weight ratio (TWR) is fundamentally derived from Newton's second law of motion, which states that the net force F on an object equals its mass m times acceleration a, or F = m a.[1] For a vehicle in vertical ascent under thrust T, the net force is T - W, where W is the weight, leading to T - W = m a. Rearranging gives a = \frac{T}{m} - g, with g as gravitational acceleration; thus, a positive acceleration (enabling ascent) requires \frac{T}{m g} > 1, or TWR > 1.[8] This derivation highlights TWR as a dimensionless measure of excess thrust beyond that needed to counteract gravity.[1] In SI units, TWR is calculated as \text{TWR} = \frac{T}{m g}, where thrust T is in newtons (N), mass m is in kilograms (kg), and g is the standard gravitational acceleration of 9.81 m/s² at sea level.[1] Equivalently, since weight W = m g in newtons, it simplifies to \text{TWR} = \frac{T}{W}.[8] In imperial units, TWR uses \text{TWR} = \frac{T}{W}, with both thrust and weight in pounds-force (lbf), yielding the same dimensionless ratio.[1] Normalization to sea-level gravity ensures comparability across vehicles, as g varies slightly with altitude or location but is standardized to 9.81 m/s² for consistency.[8] To convert between unit systems, divide thrust by weight directly in consistent units, as the ratio cancels out mass and gravity terms; for example, converting thrust from lbf to N (multiply by 4.448) and weight accordingly preserves the value.[1] Practical computation often uses static TWR at takeoff, based on maximum sea-level thrust divided by fully loaded weight, but instantaneous TWR varies during flight due to changing mass (e.g., fuel burn) or thrust levels (e.g., throttling).[1] Engineers compute instantaneous values by updating m and T in real-time via onboard sensors for performance monitoring.[8]Aircraft Applications
Jet and Turbofan Aircraft
In jet and turbofan aircraft, the thrust-to-weight ratio (TWR) plays a critical role in enabling high-speed performance, particularly during takeoff, climb, and supersonic regimes where direct engine thrust must overcome aerodynamic drag and gravitational forces.[1] These engines generate thrust through high-velocity exhaust, allowing fixed-wing designs to achieve rapid acceleration and sustained high altitudes, with TWR directly influencing the excess power available for non-level flight.[1] For vertical takeoff and landing (VTOL) jet configurations, such as those employing vectored thrust, a TWR greater than 1 is essential to achieve and maintain hover, as the engine thrust must at minimum balance the aircraft's weight in the absence of aerodynamic lift.[9] To enable acceleration from hover into forward flight or climb, the TWR typically exceeds 1.1, accounting for control margins and initial vertical momentum requirements.[9] This high TWR demand imposes significant design constraints on engine sizing and airframe integration in VTOL jets. The TWR profoundly affects climb rate and initial acceleration in both subsonic and supersonic jets, as excess thrust beyond that required for level flight converts directly into vertical velocity or speed gains.[1] In subsonic operations, a higher TWR enhances initial takeoff acceleration and service ceiling attainment by increasing the rate of climb, which is proportional to the difference between available and required thrust.[1] For supersonic jets, elevated TWR supports sustained transonic acceleration and maneuverability at high Mach numbers, where drag rises nonlinearly, allowing pilots to maintain energy states during dynamic flight phases.[1] Typical TWR values at takeoff for commercial airliners range from 0.2 to 0.4, balancing efficient cruise performance with sufficient initial thrust for safe departure under varying field lengths and environmental conditions.[10] In contrast, fighter jets often achieve TWRs of 1.0 or higher, enabling supermaneuverability and rapid response in combat scenarios.[11] Afterburning, or reheat, temporarily boosts TWR in military turbofan engines by injecting additional fuel into the exhaust for secondary combustion, potentially increasing thrust by 50-100% and elevating TWR above 1.5 for short bursts.[12] This enhancement is crucial for combat maneuvers like vertical climbs or supersonic dashes, but its use is limited to seconds or minutes due to thermal limits and fuel depletion.[12] Higher TWR designs in jet aircraft generally incur fuel consumption trade-offs, as larger engines or afterburner operation increase specific fuel consumption and induced drag, thereby reducing overall mission range.[13] For instance, prioritizing TWR for performance can elevate thrust-specific fuel consumption by 20-50% during high-power settings, necessitating careful optimization between acceleration capability and endurance.[13]Propeller-Driven Aircraft
In propeller-driven aircraft, thrust is generated by accelerating a large mass of air to a relatively low velocity using a rotating propeller, rather than expelling a small mass at high velocity as in jet engines. The effective thrust-to-weight ratio (TWR) accounts for this indirect mechanism and is calculated as TWR_eff = (η P) / (m g V), where η is the propeller efficiency (typically 0.7–0.85), P is the engine shaft power, m is the aircraft mass, g is gravitational acceleration, and V is the aircraft's forward velocity. [14] This velocity dependence means TWR decreases with increasing speed, distinguishing it from the more constant TWR in jets. [15] Typical TWR values for propeller-driven aircraft range from 0.1 to 0.3 at takeoff conditions, reflecting their reliance on airflow speed for thrust generation. [7] For example, the Cessna 172 piston-engine trainer achieves approximately 0.25 at maximum takeoff weight, based on static thrust estimates equating to about 25% of gross weight. [16] Modern turboprops like the ATR 72 exhibit around 0.21, derived from combined engine power of 4,000 shaft horsepower and a maximum takeoff weight of 48,500 pounds, assuming 80% propeller efficiency at typical liftoff velocity. [17] The TWR plays a critical role in takeoff performance, particularly for short-field operations in piston and turboprop designs, where higher ratios enable faster acceleration and reduced ground roll. [1] In piston aircraft like the Cessna 172, a TWR of 0.25 supports takeoff distances under 1,000 feet on standard runways, while lower ratios limit performance on unpaved or obstructed fields. [16] Turboprops benefit from efficient low-speed thrust, allowing shorter takeoffs than comparable jets; for instance, the ATR 72 requires about 4,500 feet at sea level, aided by its TWR facilitating rapid acceleration to rotation speed. [18] Historically, TWR in propeller aircraft has evolved with advancements in engine power and propeller design. Early biplanes like the 1903 Wright Flyer achieved around 0.16 static TWR, with 120–130 pounds of thrust from a 12-horsepower engine against a gross weight of approximately 750 pounds, sufficient for initial controlled flight but limiting climb rates to under 500 feet per minute. [19] World War I fighters such as the Sopwith Camel, with a 130-horsepower rotary engine and gross weight of 1,453 pounds, reached about 0.15 effective TWR at takeoff, enabling agile maneuvering but constraining top speeds to 115 miles per hour. [20] Modern turboprops represent an improvement to around 0.25, as seen in the ATR 72, due to higher power-to-weight ratios (up to 2.7 shaft horsepower per pound in engines) and variable-pitch propellers optimizing efficiency across speeds. [21] Propeller-driven aircraft face limitations in TWR at high altitudes and speeds, where reduced air density lowers mass flow through the propeller, decreasing thrust and efficiency. [22] At altitudes above 20,000 feet, propeller efficiency drops below 70% due to lower Reynolds numbers on blades, limiting climb rates and service ceilings for piston designs to around 15,000–18,000 feet. [23] High speeds exacerbate this, as blade tip speeds approach Mach 0.8–0.9, inducing compressibility effects, shock waves, and efficiency losses up to 20%, capping practical velocities at 400–500 miles per hour even in turboprops. [22]Rocket Applications
Launch Vehicles
In launch vehicles, the thrust-to-weight ratio (TWR) at liftoff must exceed 1 to achieve net positive acceleration, but practical designs target a minimum of 1.1 to 1.3 to overcome Earth's gravity and initial atmospheric drag while maintaining structural integrity and control margins.[24] This threshold ensures the vehicle clears the launch pad with sufficient upward acceleration, typically around 1.2 g, preventing excessive time in the dense lower atmosphere where drag losses are highest. For instance, the Saturn V rocket's first stage achieved an initial TWR of approximately 1.2, with total liftoff thrust of 7.6 million pounds-force against a gross weight of 6.35 million pounds.[25] During the ascent phase of a single stage, the TWR increases progressively as propellant is consumed, reducing the vehicle's mass while thrust remains roughly constant, which can lead to accelerating forces exceeding 3-4 g by burnout if not managed. Staging mitigates this by jettisoning empty tanks and lower stages, resetting the TWR to a controlled level for subsequent burns and optimizing overall performance by discarding non-propulsive mass. This dynamic variation is critical for multi-stage rockets pursuing Earth-to-orbit trajectories, as unchecked TWR growth could impose excessive aerodynamic and structural loads. Liquid-propellant rockets typically operate with stage TWRs of 1.2 to 2.0 at ignition, offering precise throttle control and restart capability but requiring complex turbopumps that add mass. In contrast, solid-propellant boosters can deliver higher initial TWRs, typically 2.0 to 3.0, due to their simpler, denser designs without moving parts, providing rapid thrust buildup ideal for augmenting liftoff but with limited controllability once ignited. To maintain stability during high-TWR ascent, launch vehicles employ engine gimballing and thrust vector control (TVC) systems, which pivot nozzles by several degrees to generate corrective torques against aerodynamic and gravitational perturbations. These mechanisms are essential for vehicles with TWRs near or above 1.5, where inertial forces amplify any misalignment, and hydraulic or electromechanical actuators ensure responsive steering throughout the burn.[26]In-Space Propulsion
In space propulsion systems, where gravitational forces are negligible or absent, the traditional thrust-to-weight ratio (TWR) becomes theoretically infinite since the vehicle's weight approaches zero. Instead, the relevant performance metric shifts to the thrust-to-mass ratio (T/m), which directly determines the vehicle's acceleration according to Newton's second law: a = \frac{T}{m}, where T is thrust and m is the vehicle's mass. This acceleration governs the rate of velocity change during maneuvers, orbit adjustments, or trajectory corrections in vacuum environments.[27] Low-TWR systems, such as ion thrusters, prioritize high specific impulse (Isp) over immediate thrust, resulting in equivalent TWR values typically ranging from 0.0001 to 0.01 for operational spacecraft. For example, NASA's NSTAR ion thruster on the Deep Space 1 mission produced a maximum thrust of about 92 mN with a spacecraft dry mass of approximately 486 kg, yielding an acceleration of roughly 0.00019 m/s² and an equivalent TWR (acceleration divided by standard gravity, 9.81 m/s²) of around 0.000019. These systems excel in efficient, long-duration burns for missions requiring gradual delta-v accumulation, such as deep-space trajectory corrections or station-keeping, where fuel efficiency outweighs the need for rapid response.[28][29] In contrast, high-TWR chemical propulsion systems used in upper stages achieve equivalent TWRs of 0.5 to 1.0, enabling swift acceleration for time-sensitive operations like orbit insertion or interplanetary injection. The Centaur upper stage, powered by two RL10 engines delivering a combined thrust of 146.8 kN at ignition with a full mass of about 23,130 kg, provides an initial acceleration of approximately 6.35 m/s², corresponding to a TWR equivalent of 0.65. Such capabilities support rapid maneuvers that minimize exposure to radiation or thermal stresses during short burns.[30] The choice between high- and low-TWR systems profoundly influences mission design: high-TWR chemical rockets facilitate quick, high-delta-v maneuvers essential for precise orbit raising or planetary escapes, reducing overall mission duration but consuming more propellant, while low-TWR electric systems like ion thrusters enable fuel-efficient station-keeping or slow spiral trajectories for extended operations, such as maintaining geostationary orbits or enabling low-thrust transfers to distant targets.[31] Emerging nuclear thermal propulsion (NTP) technologies aim to bridge this gap by targeting TWR equivalents greater than 2 in space, offering both higher thrust than electric systems and improved efficiency over chemical ones. NASA's Pewee-class NTP engine, derived from historical NERVA designs, achieves a TWR of about 3.5 with a specific impulse of 875–950 seconds, potentially enabling faster Mars transits in the 2030s by reducing travel times and propellant mass compared to conventional upper stages.[32]Performance and Design Implications
Acceleration and Maneuverability
The thrust-to-weight ratio (TWR) fundamentally governs a vehicle's linear acceleration capabilities, particularly in scenarios where thrust opposes gravitational forces. For vertical motion neglecting drag, the net upward acceleration a is derived from Newton's second law as a = g (TWR - 1), where g is the standard gravitational acceleration (approximately 9.8 m/s²), and this holds when TWR exceeds 1 to produce positive acceleration.[1] This relationship highlights how excess thrust beyond the vehicle's weight enables powered ascent, with the acceleration scaling linearly with the excess TWR. Threshold values of TWR delineate critical performance boundaries. At TWR = 1, thrust exactly balances weight, allowing stationary hover in a gravitational field without net motion.[1] Conversely, when TWR < 1, insufficient thrust prevents sustained vertical climb, confining the vehicle to gliding paths in atmospheric conditions or suborbital trajectories in space, where gravitational forces dominate without adequate propulsion to achieve orbital velocity.[1] Beyond linear motion, elevated TWR enhances angular maneuverability by providing the excess power needed for rapid changes in direction. In high-performance applications such as fighters and missiles, higher TWR facilitates tighter turning radii and quicker pitch rates, as the surplus thrust supports sustained forces during dynamic maneuvers without excessive speed loss.[33] This link arises because greater excess thrust correlates with improved energy retention and control authority in multi-dimensional flight regimes.[34] Thrust vectoring extends these effects into multi-axis control by redirecting thrust components, effectively amplifying the TWR in off-axis directions to generate control moments. This technique enhances stability and responsiveness, allowing vehicles to execute precise attitude adjustments that would otherwise require unattainable conventional forces.[35] In orbital mechanics contexts, TWR influences overall performance through its role in minimizing gravity losses during simulated ascents. Higher TWR enables shorter burn durations to impart the necessary delta-v, reducing the integrated gravitational drag over time and thereby optimizing the effective velocity change available for reaching orbit.[36]Design Considerations
In vehicle design, achieving a high thrust-to-weight ratio (TWR) must be balanced against structural integrity to prevent catastrophic failures, such as engine-out scenarios or excessive airframe stress during high-acceleration phases. In rockets, the thrust structure, which mounts engines and transmits loads to the body, often constitutes a significant portion of the total structural mass, with fractions typically ranging from 0.05 to 0.15 depending on vehicle type; for reusable boosters, this can be higher, requiring robust materials to withstand dynamic pressures around 20-40 kPa at maximum dynamic pressure (max-q) while minimizing mass to preserve payload fraction.[37] Similarly, in aircraft, pursuing higher TWR through elevated turbine inlet temperatures enhances acceleration but demands advanced alloys and cooling systems to maintain airframe durability under thermal and vibrational loads. These trade-offs prioritize lightweight composites and optimized load paths, ensuring the vehicle can handle thrust-induced stresses without compromising safety margins. To mitigate risks from high TWR, multi-engine redundancy via clustering is a standard strategy, distributing thrust across multiple units to achieve the desired ratio while enabling continued operation despite a single-point failure. In rocket propulsion, designs like the Saturn V's first stage, with five clustered engines, incorporate engine-out capabilities through redundant actuators, sensors, and propellant feed systems, allowing mission abort or continuation by throttling remaining engines; this approach, detailed in liquid propellant engine guidelines, uses gimbaled clusters with precise alignment to maintain stability and vector control. Aircraft employ analogous redundancy, such as selective engine thrust modulation in multi-engine transports, where asymmetric thrust compensates for failures, preserving controllability and TWR during critical phases like takeoff; this is facilitated by rapid-response turbojets and backup flight control integration, reducing the impact of thrust loss on overall performance. Scalability poses significant challenges in maintaining TWR for larger vehicles, as increased mass demands higher total thrust, complicating engine integration and structural scaling. For launch vehicles, larger solid-propellant systems generally achieve better mass efficiency than smaller ones due to improved structural proportions, but liquid systems face consistent challenges in scaling, leading to prolonged burn times and higher gravity losses in massive configurations; this necessitates advanced clustering or staging to counteract inefficiencies, with small vehicles (<1 ton) suffering from elevated drag due to higher surface-area-to-mass ratios, requiring steeper trajectories and up to 10 km/s delta-v. Optimizing TWR for cost and efficiency involves tailoring it to specific mission profiles, where reusable systems often favor lower peak values to extend hardware life and minimize operational expenses. In reusable rockets, all-propulsive return trajectories reserve approximately 20-30% of first-stage propellant for landing, yielding liftoff TWRs around 1.2-1.4 and lower values during descent, which reduces payload to 50-60% of expendable counterparts but enables cost savings through rapid turnaround. This approach balances reusability penalties—like added inert mass for thermal protection—with efficiency gains in low Earth orbit missions, prioritizing controlled descents over maximum ascent thrust. As of 2025, ongoing developments in vehicles like SpaceX's Starship have further optimized these trade-offs, achieving higher reusability with reduced structural mass fractions through advanced materials and design iterations.[38] Emerging trends in electric and hybrid propulsion promise substantial TWR enhancements by 2030, driven by higher power densities in motors and batteries. Hybrid-electric aircraft concepts target electric motor specific powers of 13–16 kW/kg, enabling distributed propulsion architectures that reduce overall system weight and improve effective TWR for regional flights; for instance, 2030 single-aisle designs optimize TWR to 0.28–0.36 through electrified powertrains, yielding 20–21% fuel burn reductions via lighter components and boundary layer ingestion. These advancements, focused on urban air mobility and sustainable aviation, will facilitate higher thrust efficiency without proportional mass increases, supporting broader adoption in both atmospheric and in-space applications.Examples
Aircraft Instances
The Boeing 747, a quintessential commercial airliner, exhibits a takeoff thrust-to-weight ratio (TWR) of approximately 0.26 at maximum takeoff weight, enabling an initial horizontal acceleration of roughly 2.5 m/s² during the takeoff roll, which facilitates safe departure from runways despite its substantial mass.[39] This value reflects the balance between the four high-bypass turbofan engines' collective output and the aircraft's loaded weight, prioritizing fuel efficiency and range over rapid climb rates typical in military designs.[39] In contrast, the Lockheed Martin F-22 Raptor, a fifth-generation stealth fighter, achieves a TWR of approximately 1.08 with afterburners engaged at a typical combat weight of around 64,840 pounds (supported by 70,000 lbf total thrust from its twin F119 engines), enabling sustained supercruise at Mach 1.5 without afterburner and exceptional vertical performance.[40] This ratio underscores the F-22's design emphasis on agility, allowing it to outperform adversaries in dogfights and rapid intercepts.[41] Historically, the Wright Flyer of 1903 represented the nascent era of powered flight with an estimated TWR of 0.16, derived from its 12-horsepower engine producing about 120-130 pounds of propeller thrust against a gross weight of roughly 750 pounds, which constrained its top speed to approximately 12 m/s and necessitated rail-assisted launches into headwinds.[19] For vertical/short takeoff and landing (V/STOL) operations, the Hawker Siddeley Harrier Jump Jet demonstrates a TWR of about 1.05 in a lightly loaded VTOL configuration (with 21,500-23,500 lbf thrust from its Pegasus engine at weights up to around 22,000 pounds), permitting short takeoff rolls of under 500 feet while maintaining hover capability.[42] The progression of TWR in aircraft design illustrates advancements in propulsion technology, from the low ratios of early piston-engine pioneers to the supermaneuverable profiles of modern jets, driven by improvements in engine efficiency and materials. For example, the Lockheed Martin F-35 Lightning II achieves a TWR of approximately 0.87 at combat weight (as of 2025), balancing stealth and multirole capabilities.| Era | Representative Type/Example | Approximate TWR | Key Context |
|---|---|---|---|
| Early Piston (1900s) | Wright Flyer | 0.16 | Limited speed and required assistance for takeoff; propeller thrust barely exceeded weight.[19] |
| Piston (WWII) | P-51 Mustang | 0.25 | Enabled agile dogfighting with power-to-weight equivalent supporting dives up to 500 mph. |
| Early Jet (1950s) | F-86 Sabre | 0.50 | Marked transition to transonic speeds but sub-1.0 ratio limited vertical climbs.[43] |
| Modern Jet (2000s+) | F-22 Raptor (afterburner) | 1.08+ | Supports supercruise and post-stall maneuvers, exceeding unity for vertical takeoffs.[40] |
Rocket Instances
The SpaceX Falcon 9 first stage exemplifies a modern launch vehicle booster with a thrust-to-weight ratio (TWR) of approximately 1.3 at liftoff, enabling efficient ascent and the achievement of accelerations up to 10g during flight phases where mass decreases significantly.[44] Solid rocket boosters, such as those used on the Space Shuttle, demonstrate high initial TWR values around 2.2 at ignition, providing the majority of liftoff thrust for the stack, though their fixed-burn profile required careful integration with liquid engines to maintain stability without throttling capability.[45] In contrast, upper stages like the Centaur, powered by RL10 engines, operate with a lower TWR of approximately 0.7 in vacuum conditions, prioritizing high specific impulse for precise orbital insertion and trajectory adjustments over rapid acceleration.[30] Historically, the German V-2 rocket achieved a TWR of about 2.1, which was sufficient to enable the first successful suborbital ballistic flight reaching approximately 80 km altitude in 1944, marking a pivotal advancement in rocketry despite its relatively modest performance by modern standards.[46] The following table compares TWR values for chemical propulsion in launch vehicles versus electric propulsion in deep-space missions, illustrated by the Dawn spacecraft's ion system with an equivalent TWR of approximately 0.0003, highlighting the trade-off between high-thrust chemical stages for escape and low-thrust electric systems for efficient interplanetary cruise. Modern examples include the SpaceX Starship, with a projected TWR of about 1.5 at liftoff for its Super Heavy booster (as of 2025 testing).[38]| Propulsion Type | Example Mission/Vehicle | Approximate TWR | Mission Impact |
|---|---|---|---|
| Chemical (Liquid) | SpaceX Falcon 9 First Stage | 1.3 | Rapid Earth escape and orbit insertion |
| Chemical (Solid) | Space Shuttle SRB | 2.2 | Dominant liftoff thrust for heavy-lift stacks |
| Chemical (Upper Stage) | Centaur RL10 | 0.7 | Precise vacuum maneuvers and circularization |
| Electric (Ion) | NASA Dawn Spacecraft | 0.0003 | Extended low-acceleration thrusting for asteroid rendezvous over years |