Gravity turn
Encyclopedia
A gravity turn or zero-lift turn is a maneuver (see trajectory optimization
) used in launching a spacecraft into, or descending from, an orbit
around a celestial body such as a planet
or a moon
. This launch trajectory
offers two main advantages over a thrust-controlled trajectory where the rocket's
own thrust
steers the vehicle. First, any thrust used to change the ship's direction does not accelerate the vehicle into orbit. This loss can be reduced by using gravity to steer the vehicle onto its desired trajectory. Second, and more importantly, because gravity does the steering during the initial ascent phase the vehicle can maintain low or even zero angle of attack
. This minimizes transverse aerodynamic stress
on the launch vehicle, allowing for a lighter launch vehicle.
The term gravity turn can also refer to the use of a planet's gravity to change a spacecraft's direction. When used in this context it is similar to a gravitational slingshot
; the difference is that a gravitational slingshot often increases or decreases spacecraft velocity and changes direction while the gravity turn only changes direction.
which launch vertically. The rocket begins by flying straight up, gaining both vertical speed and altitude. During this portion of the launch gravity acts directly against the thrust of the rocket, lowering its vertical acceleration. Losses associated with this slowing are known as gravity drag
, and can be minimized by executing the next phase of the launch, the pitch over maneuver, as soon as possible. The pitch over should also be carried out while the vertical velocity is small to avoid large aerodynamic loads on the vehicle during the maneuver.
The pitch over maneuver consists of the rocket gimbaling its engine slightly to direct some of its thrust to one side. This force creates a net torque on the ship, turning it so that it no longer points vertically. The pitch over angle varies with the launch vehicle and is included in the rocket's initial guidance system, for some vehicles it is only a few degrees while other vehicles use relatively large angles (a few tens of degrees). After the pitch over is complete the engines are reset to point straight down the axis of the rocket again. This small steering maneuver is the only time during an ideal gravity turn ascent that thrust must be used for purposes of steering. This pitch over maneuver serves two purposes. First, it turns the rocket slightly so that its flight path is no longer vertical, and second, it places the rocket on the correct heading
for its ascent to orbit. After the pitch over the rocket's angle of attack is adjusted to zero for the remainder of its climb to orbit. This zeroing of the angle of attack reduces lateral aerodynamic loads and produces negligible lift force during the ascent.
like a thrown ball, leveling off and then falling back to the ground. The rocket is producing thrust though, and rather than leveling off and then descending again, by the time the rocket levels off it has gained sufficient altitude and velocity to place it in a stable orbit.
If the rocket is a multi-stage system where stages fire sequentially, the rocket's ascent burn may not be continuous. Obviously some time must be allowed for stage separation and engine ignition between each successive stage, but some rocket designs call for extra free-flight time between stages. This is particularly useful in very high thrust rockets where if the engines were fired continuously the rocket would run out of fuel before leveling off and reaching a stable orbit above the atmosphere. The technique is also useful when launching from a planet with a thick atmosphere, such as the Earth. Since gravity turns the flight path during free flight the rocket can use a smaller initial pitch over angle, giving it higher vertical velocity, and taking it out of the atmosphere more quickly. This reduces both aerodynamic drag as well as aerodynamic stress during launch. Then later during the flight the rocket coasts between stage firings allowing it to level off above the atmosphere so when the engine fires again, at zero angle of attack, the thrust accelerates the ship horizontally, inserting it into orbit.
, a powered descent with a gravity turn is a good alternative. The Apollo lunar module used a slightly modified gravity turn to land from lunar orbit. This was essentially a launch in reverse except that a landing spacecraft is lightest at the surface while a spacecraft being launched is heaviest at the surface. A computer program called Lander that simulated gravity turn landings applied this concept by simulating a gravity turn launch with a negative mass flow rate, i.e. the propellant tanks filled during the rocket burn. The idea of using a gravity turn maneuver to land a vehicle was originally developed for the Lunar Surveyor
landings, although Surveyor made a direct approach to the surface without first going into lunar orbit.
, lowering its point of periapsis to near the surface of the body to be landed on. If the craft is landing on a planet with an atmosphere such as Mars
the deorbit burn will only lower periapsis into the upper layers of the atmosphere, rather than just above the surface as on an airless body. After the deorbit burn is complete the vehicle can either coast until it is nearer to its landing site or continue firing its engine while maintaining zero angle of attack. For a planet with an atmosphere the coast portion of the trip includes reentry through the atmosphere as well.
After the coast and possible reentry the vehicle jettisons any no longer necessary heat shield
s and/or parachutes in preparation for the final landing burn. If the atmosphere is thick enough it can be used to slow the vehicle a lot, thus saving on fuel. In this case a gravity turn is not the optimal entry trajectory but it does allow for approximation of the true delta-v
required. In the case where there is no atmosphere however the landing vehicle must provide the full delta-v necessary to land safely on the surface.
, the ability to point the rocket in a desired direction, and guidance
, the determination of what direction a rocket should be pointed to reach a given target. The desired target can either be a location on the ground, as in the case of a ballistic missile
, or a particular orbit, as in the case of a launch vehicle.
s, is almost always employed on modern rockets. The British
satellite launcher Black Arrow
was an example of a rocket that flew a preprogrammed pitch schedule, making no attempt to correct for errors in its trajectory, while the Apollo-Saturn rockets used "closed loop" inertial guidance after the gravity turn through the atmosphere.
The initial pitch program is an open-loop
system subject to errors from winds, thrust variations, etc. To maintain zero angle of attack during atmospheric flight, these errors are not corrected until reaching space. Then a more sophisticated closed-loop guidance program can take over to correct trajectory deviations and attain the desired orbit. In the Apollo missions, the transition to closed-loop guidance took place early in second stage flight after maintaining a fixed inertial attitude while jettisoning the first stage and interstage ring. Because the upper stages of a rocket operate in a near vacuum, fins are ineffective. Steering relies entirely on engine gimballing and a reaction control system
.
on an airless body will be assumed. The lander begins in a circular orbit docked to the command module. After separation from the command module the lander performs a retrograde burn to lower its periapsis to just above the surface. It then coasts to periapsis where the engine is restarted to perform the gravity turn descent. It has been shown that in this situation guidance can be achieved by maintaining a constant angle between the thrust vector and the line of sight to the orbiting command module. This simple guidance algorithm builds on a previous study which investigated the use of various visual guidance cues including the uprange horizon, the downrange horizon, the desired landing site, and the orbiting command module. The study concluded that using the command module provides the best visual reference, as it maintains a near constant visual separation from an ideal gravity turn until the landing is almost complete. Because the vehicle is landing in a vacuum, aerodynamic control surfaces are useless. Therefore a system such as a gimballing main engine, a reaction control system, or possibly a control moment gyroscope
must be used for attitude control.
A variant of this maneuver, the free return trajectory
allows the spacecraft to depart from a planet, circle another planet once, and return to the starting planet using propulsion only during the initial departure burn. Although in theory it is possible to execute a perfect free return trajectory, in practice small correction burns are often necessary during the flight. Even though it does not require a burn for the return trip, other return trajectory types, such as an aerodynamic turn, can result in a lower total delta-v for the mission.
is a vector whose magnitude is a function of time and whose direction can be varied at will. Under these assumptions the differential equation of motion is given by:
Here
is a unit vector in the vertical direction and 
is the instantaneous vehicle mass. By constraining the thrust vector to point parallel to the velocity and separating the equation of motion into components parallel to 
and those perpendicular to 
we arrive at the following system:
Here the current thrust to weight ratio has been denoted by
and the current angle between the velocity vector and the vertical by 
. This results in a coupled system of equations which can be integrated to obtain the trajectory. However, for all but the simplest case of constant 
over the entire flight, the equations cannot be solved analytically
and must be integrated numerically
.
Trajectory optimization
Trajectory optimization is the process of designing a trajectory that minimizes or maximizes some measure of performance within prescribed constraint boundaries...
) used in launching a spacecraft into, or descending from, an orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
around a celestial body such as a planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
or a moon
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
. This launch trajectory
Trajectory
A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...
offers two main advantages over a thrust-controlled trajectory where the rocket's
Rocket
A rocket is a missile, spacecraft, aircraft or other vehicle which obtains thrust from a rocket engine. In all rockets, the exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction...
own thrust
Thrust
Thrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....
steers the vehicle. First, any thrust used to change the ship's direction does not accelerate the vehicle into orbit. This loss can be reduced by using gravity to steer the vehicle onto its desired trajectory. Second, and more importantly, because gravity does the steering during the initial ascent phase the vehicle can maintain low or even zero angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
. This minimizes transverse aerodynamic stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
on the launch vehicle, allowing for a lighter launch vehicle.
The term gravity turn can also refer to the use of a planet's gravity to change a spacecraft's direction. When used in this context it is similar to a gravitational slingshot
Gravitational slingshot
In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement and gravity of a planet or other celestial body to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense...
; the difference is that a gravitational slingshot often increases or decreases spacecraft velocity and changes direction while the gravity turn only changes direction.
Vertical climb
The gravity turn is commonly utilized with launch vehicles such as a rocket or the Space ShuttleSpace Shuttle
The Space Shuttle was a manned orbital rocket and spacecraft system operated by NASA on 135 missions from 1981 to 2011. The system combined rocket launch, orbital spacecraft, and re-entry spaceplane with modular add-ons...
which launch vertically. The rocket begins by flying straight up, gaining both vertical speed and altitude. During this portion of the launch gravity acts directly against the thrust of the rocket, lowering its vertical acceleration. Losses associated with this slowing are known as gravity drag
Gravity drag
In astrodynamics and rocketry, gravity drag is a measure of the loss in the net performance of a rocket while it is thrusting in a gravitational field...
, and can be minimized by executing the next phase of the launch, the pitch over maneuver, as soon as possible. The pitch over should also be carried out while the vertical velocity is small to avoid large aerodynamic loads on the vehicle during the maneuver.
The pitch over maneuver consists of the rocket gimbaling its engine slightly to direct some of its thrust to one side. This force creates a net torque on the ship, turning it so that it no longer points vertically. The pitch over angle varies with the launch vehicle and is included in the rocket's initial guidance system, for some vehicles it is only a few degrees while other vehicles use relatively large angles (a few tens of degrees). After the pitch over is complete the engines are reset to point straight down the axis of the rocket again. This small steering maneuver is the only time during an ideal gravity turn ascent that thrust must be used for purposes of steering. This pitch over maneuver serves two purposes. First, it turns the rocket slightly so that its flight path is no longer vertical, and second, it places the rocket on the correct heading
Course (navigation)
In navigation, a vehicle's course is the angle that the intended path of the vehicle makes with a fixed reference object . Typically course is measured in degrees from 0° clockwise to 360° in compass convention . Course is customarily expressed in three digits, using preliminary zeros if needed,...
for its ascent to orbit. After the pitch over the rocket's angle of attack is adjusted to zero for the remainder of its climb to orbit. This zeroing of the angle of attack reduces lateral aerodynamic loads and produces negligible lift force during the ascent.
Downrange acceleration
After the pitch over, the rocket's flight path is no longer completely vertical so gravity acts to turn the flight path back towards the ground. If the rocket were not producing thrust the flight path would be a simple parabolaParabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
like a thrown ball, leveling off and then falling back to the ground. The rocket is producing thrust though, and rather than leveling off and then descending again, by the time the rocket levels off it has gained sufficient altitude and velocity to place it in a stable orbit.
If the rocket is a multi-stage system where stages fire sequentially, the rocket's ascent burn may not be continuous. Obviously some time must be allowed for stage separation and engine ignition between each successive stage, but some rocket designs call for extra free-flight time between stages. This is particularly useful in very high thrust rockets where if the engines were fired continuously the rocket would run out of fuel before leveling off and reaching a stable orbit above the atmosphere. The technique is also useful when launching from a planet with a thick atmosphere, such as the Earth. Since gravity turns the flight path during free flight the rocket can use a smaller initial pitch over angle, giving it higher vertical velocity, and taking it out of the atmosphere more quickly. This reduces both aerodynamic drag as well as aerodynamic stress during launch. Then later during the flight the rocket coasts between stage firings allowing it to level off above the atmosphere so when the engine fires again, at zero angle of attack, the thrust accelerates the ship horizontally, inserting it into orbit.
Descent and landing procedure
Because heat shields and parachutes cannot be used to land on an airless body such as the MoonMoon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
, a powered descent with a gravity turn is a good alternative. The Apollo lunar module used a slightly modified gravity turn to land from lunar orbit. This was essentially a launch in reverse except that a landing spacecraft is lightest at the surface while a spacecraft being launched is heaviest at the surface. A computer program called Lander that simulated gravity turn landings applied this concept by simulating a gravity turn launch with a negative mass flow rate, i.e. the propellant tanks filled during the rocket burn. The idea of using a gravity turn maneuver to land a vehicle was originally developed for the Lunar Surveyor
Surveyor program
The Surveyor Program was a NASA program that, from 1966 through 1968, sent seven robotic spacecraft to the surface of the Moon. Its primary goal was to demonstrate the feasibility of soft landings on the Moon...
landings, although Surveyor made a direct approach to the surface without first going into lunar orbit.
Deorbit and reentry
The vehicle begins by orienting for a retrograde burn to reduce its orbital velocityOrbital velocity
Orbital velocity can refer to the following:* The orbital speed of a body in a gravitational field.* The velocity of particles due to wave motion, in particular in wind waves....
, lowering its point of periapsis to near the surface of the body to be landed on. If the craft is landing on a planet with an atmosphere such as Mars
Mars
Mars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
the deorbit burn will only lower periapsis into the upper layers of the atmosphere, rather than just above the surface as on an airless body. After the deorbit burn is complete the vehicle can either coast until it is nearer to its landing site or continue firing its engine while maintaining zero angle of attack. For a planet with an atmosphere the coast portion of the trip includes reentry through the atmosphere as well.
After the coast and possible reentry the vehicle jettisons any no longer necessary heat shield
Heat shield
A heat shield is designed to shield a substance from absorbing excessive heat from an outside source by either dissipating, reflecting or simply absorbing the heat...
s and/or parachutes in preparation for the final landing burn. If the atmosphere is thick enough it can be used to slow the vehicle a lot, thus saving on fuel. In this case a gravity turn is not the optimal entry trajectory but it does allow for approximation of the true delta-v
Delta-v
In astrodynamics a Δv or delta-v is a scalar which takes units of speed. It is a measure of the amount of "effort" that is needed to change from one trajectory to another by making an orbital maneuver....
required. In the case where there is no atmosphere however the landing vehicle must provide the full delta-v necessary to land safely on the surface.
Landing
If it is not already properly oriented, the vehicle lines up its engines to fire directly opposite its current surface velocity vector, which at this point is either parallel to the ground or only slightly vertical, as shown to the left. The vehicle then fires its landing engine to slow down for landing. As the vehicle loses horizontal velocity the gravity of the body to be landed on will begin pulling the trajectory closer and closer to a vertical descent. In an ideal maneuver on a perfectly spherical body the vehicle could reach zero horizontal velocity, zero vertical velocity, and zero altitude all at the same moment, landing safely on the surface. However due to rocks and uneven surface terrain the vehicle usually picks up a few degrees of angle of attack near the end of the maneuver to zero its horizontal velocity just above the surface. This process is the mirror image if the pitch over maneuver used in the launch procedure and allows the vehicle to hover straight down, landing gently on the surface.Guidance and control
The steering of a rocket's course during its flight is divided into two separate components; controlControl system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
, the ability to point the rocket in a desired direction, and guidance
Guidance system
A guidance system is a device or group of devices used to navigate a ship, aircraft, missile, rocket, satellite, or other craft. Typically, this refers to a system that navigates without direct or continuous human control...
, the determination of what direction a rocket should be pointed to reach a given target. The desired target can either be a location on the ground, as in the case of a ballistic missile
Ballistic missile
A ballistic missile is a missile that follows a sub-orbital ballistic flightpath with the objective of delivering one or more warheads to a predetermined target. The missile is only guided during the relatively brief initial powered phase of flight and its course is subsequently governed by the...
, or a particular orbit, as in the case of a launch vehicle.
Launch
The gravity turn trajectory is most commonly used during early ascent. The guidance program is a precalculated lookup table of pitch vs time. Control is done with engine gimballing and/or aerodynamic control surfaces. The pitch program maintains a zero angle of attack (the definition of a gravity turn) until the vacuum of space is reached, thus minimizing lateral aerodynamic loads on the vehicle. (Excessive aerodynamic loads can quickly destroy the vehicle.) Although the preprogrammed pitch schedule is adequate for some applications, an adaptive inertial guidance system that determines location, orientation and velocity with accelerometers and gyroscopeGyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...
s, is almost always employed on modern rockets. The British
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
satellite launcher Black Arrow
Black Arrow
Black Arrow, officially capitalised BLACK ARROW, was a British satellite carrier rocket. Developed during the 1960s, it was used for four launches between 1969 and 1971...
was an example of a rocket that flew a preprogrammed pitch schedule, making no attempt to correct for errors in its trajectory, while the Apollo-Saturn rockets used "closed loop" inertial guidance after the gravity turn through the atmosphere.
The initial pitch program is an open-loop
Open-loop controller
An open-loop controller, also called a non-feedback controller, is a type of controller that computes its input into a system using only the current state and its model of the system....
system subject to errors from winds, thrust variations, etc. To maintain zero angle of attack during atmospheric flight, these errors are not corrected until reaching space. Then a more sophisticated closed-loop guidance program can take over to correct trajectory deviations and attain the desired orbit. In the Apollo missions, the transition to closed-loop guidance took place early in second stage flight after maintaining a fixed inertial attitude while jettisoning the first stage and interstage ring. Because the upper stages of a rocket operate in a near vacuum, fins are ineffective. Steering relies entirely on engine gimballing and a reaction control system
Reaction control system
A reaction control system is a subsystem of a spacecraft whose purpose is attitude control and steering by the use of thrusters. An RCS system is capable of providing small amounts of thrust in any desired direction or combination of directions. An RCS is also capable of providing torque to allow...
.
Landing
To serve as an example of how the gravity turn can be used for a powered landing, an Apollo type landerApollo Lunar Module
The Apollo Lunar Module was the lander portion of the Apollo spacecraft built for the US Apollo program by Grumman to carry a crew of two from lunar orbit to the surface and back...
on an airless body will be assumed. The lander begins in a circular orbit docked to the command module. After separation from the command module the lander performs a retrograde burn to lower its periapsis to just above the surface. It then coasts to periapsis where the engine is restarted to perform the gravity turn descent. It has been shown that in this situation guidance can be achieved by maintaining a constant angle between the thrust vector and the line of sight to the orbiting command module. This simple guidance algorithm builds on a previous study which investigated the use of various visual guidance cues including the uprange horizon, the downrange horizon, the desired landing site, and the orbiting command module. The study concluded that using the command module provides the best visual reference, as it maintains a near constant visual separation from an ideal gravity turn until the landing is almost complete. Because the vehicle is landing in a vacuum, aerodynamic control surfaces are useless. Therefore a system such as a gimballing main engine, a reaction control system, or possibly a control moment gyroscope
Control moment gyroscope
A control momentum gyroscope is an attitude control device generally used in spacecraft attitude control systems. A CMG consists of a spinning rotor and one or more motorized gimbals that tilt the rotor’s angular momentum. As the rotor tilts, the changing angular momentum causes a gyroscopic...
must be used for attitude control.
Limitations
Although gravity turn trajectories use minimal steering thrust they are not always the most efficient possible launch or landing procedure. Several things can affect the gravity turn procedure making it less efficient or even impossible due to the design limitations of the launch vehicle. A brief summary of factors affecting the turn is given below.- AtmosphereAtmosphereAn atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
— In order to minimize gravity dragGravity dragIn astrodynamics and rocketry, gravity drag is a measure of the loss in the net performance of a rocket while it is thrusting in a gravitational field...
the vehicle should begin gaining horizontal speed as soon as possible. On an airless body such as the Moon this presents no problem, however on a planet with a dense atmosphere this is not possible. A trade off exists between flying higher before starting downrange acceleration, thus increasing gravity drag losses; or starting downrange acceleration earlier, reducing gravity drag but increasing the aerodynamic drag experienced during launch.
- Maximum dynamic pressureMax QIn aerospace engineering, the maximum dynamic pressure, often referred to as maximum Q or max Q, is the point at which aerodynamic stress on a vehicle in atmospheric flight is maximized...
— Another effect related to the planet's atmosphere is the maximum dynamic pressure exerted on the launch vehicle during the launch. Dynamic pressure is related to both the atmospheric density and the vehicle's speed through the atmosphere. Just after liftoff the vehicle is gaining speed and increasing dynamic pressure faster than the reduction in atmospheric density can decrease the dynamic pressure. This causes the dynamic pressure exerted on the vehicle to increase until the two rates are equal. This is known as the point of maximum dynamic pressure (abbreviated "max QMax QIn aerospace engineering, the maximum dynamic pressure, often referred to as maximum Q or max Q, is the point at which aerodynamic stress on a vehicle in atmospheric flight is maximized...
"), and the launch vehicle must be built to withstand this amount of stress during launch. As before a trade off exists between gravity drag from flying higher first to avoid the thicker atmosphere when accelerating; or accelerating more at lower altitude, resulting in a heavier launch vehicle because of a higher maximum dynamic pressure experienced on launch.
- Maximum engine thrustThrustThrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....
— The maximum thrust the rocket engine can produce affects several aspects of the gravity turn procedure. First before the pitch over maneuver the vehicle must be capable of not only overcoming the force of gravity but accelerating upwards. The more acceleration the vehicle has beyond the acceleration of gravity the quicker vertical speed can be obtained allowing for lower gravity drag in the initial launch phase. When the pitch over is executed the vehicle begins its downrange acceleration phase; engine thrust affects this phase as well. Higher thrust allows for a faster acceleration to orbital velocity as well. By reducing this time the rocket can level off sooner; further reducing gravity drag losses. Although higher thrust can make the launch more efficient, accelerating too much low in the atmosphere increases the maximum dynamic pressure. This can be alleviated by throttling the engines back during the beginning of downrange acceleration until the vehicle has climbed higher. However, with solid fuel rockets this may not be possible.
- Maximum payload acceleration — Another limitation related to engine thrust is the maximum acceleration that can be safely sustained by the crew and/or the payload. Near main engine cut off (MECO) when the launch vehicle has consumed most of its fuel it will be much lighter than it was at launch. If the engines are still producing the same amount of thrust the acceleration will grow as a result of the decreasing vehicle mass. If this acceleration is not kept in check by throttling back the engines injury to the crew or damage to the payload could occur. This forces the vehicle to spend more time gaining horizontal velocity, increasing gravity drag.
Use in orbital redirection
For spacecraft missions where large changes in the direction of flight are necessary, direct propulsion by the spacecraft may not be feasible due to the large delta-v requirement. In these cases it may be possible to perform a flyby of a nearby planet or moon, using its gravitational attraction to alter the ship's direction of flight. Although this maneuver is very similar to the gravitational slingshot it differs in that a slingshot often implies a change in both speed and direction whereas the gravity turn only changes the direction of flight.A variant of this maneuver, the free return trajectory
Free return trajectory
A free return trajectory is one of a very small sub-class of trajectories in which the trajectory of a satellite traveling away from a primary body is modified by the presence of a secondary body causing the satellite to return to the primary body...
allows the spacecraft to depart from a planet, circle another planet once, and return to the starting planet using propulsion only during the initial departure burn. Although in theory it is possible to execute a perfect free return trajectory, in practice small correction burns are often necessary during the flight. Even though it does not require a burn for the return trip, other return trajectory types, such as an aerodynamic turn, can result in a lower total delta-v for the mission.
Use in spaceflight
Many spaceflight missions have utilized the gravity turn, either directly or in a modified form, to carry out their missions. What follows is a short list of various mission that have used this procedure.- Surveyor programSurveyor programThe Surveyor Program was a NASA program that, from 1966 through 1968, sent seven robotic spacecraft to the surface of the Moon. Its primary goal was to demonstrate the feasibility of soft landings on the Moon...
— A precursor to the Apollo Program, the Surveyor Program's primary mission objective was to develop the ability to perform soft landings on the surface of the moon, through the use of an automated descent and landing program built into the lander. Although the landing procedure can be classified as a gravity turn descent, it differs from the technique most commonly employed in that it was shot from the Earth directly to the lunar surface, rather than first orbiting the moon as the Apollo landers did. Because of this the descent path was nearly vertical, although some "turning" was done by gravity during the landing.
- Apollo program — Launches of the Saturn VSaturn VThe Saturn V was an American human-rated expendable rocket used by NASA's Apollo and Skylab programs from 1967 until 1973. A multistage liquid-fueled launch vehicle, NASA launched 13 Saturn Vs from the Kennedy Space Center, Florida with no loss of crew or payload...
rocket during the Apollo program were carried out using a gravity turn in order to minimize lateral stress on the rocket. At the other end of their journey, the lunar landers utilized a gravity turn landing and ascent from the moon.
- Mariner 10Mariner 10Mariner 10 was an American robotic space probe launched by NASA on November 3, 1973, to fly by the planets Mercury and Venus. It was launched approximately two years after Mariner 9 and was the last spacecraft in the Mariner program...
— The Mariner 10 mission used a gravity assist from the planet Venus to travel to Mercury. In 1970, three years before its launch, Giuseppe Colombo noticed that because the spacecraft's orbit around the Sun after the encounter with Mercury was very close to twice the orbital period of Mercury. By properly orienting the first flyby of Mercury the spacecraft underwent a gravity turn which allowed it to make a second flyby of the planet.
- Ulysses — The Ulysses probe utilized a gravity turn around JupiterJupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
to change the inclination of its orbit around the sun. This was done because the delta-v required to launch into a polar orbit around the sun was greater than the capability of any existing rocket. The spacecraft left Earth, arriving at Jupiter slightly "below" it; this caused Jupiter's gravity to incline the orbit so the probe would pass over the Sun's "north" pole.
Mathematical description
The simplest case of the gravity turn trajectory is that which describes a point mass vehicle, in a uniform gravitational field, neglecting air resistance. The thrust force is a vector whose magnitude is a function of time and whose direction can be varied at will. Under these assumptions the differential equation of motion is given by:Here
is a unit vector in the vertical direction and is the instantaneous vehicle mass. By constraining the thrust vector to point parallel to the velocity and separating the equation of motion into components parallel to and those perpendicular to we arrive at the following system:Here the current thrust to weight ratio has been denoted by
and the current angle between the velocity vector and the vertical by . This results in a coupled system of equations which can be integrated to obtain the trajectory. However, for all but the simplest case of constant over the entire flight, the equations cannot be solved analyticallyClosed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
and must be integrated numerically
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
.