Escape velocity

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In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account the fact that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.

Contents

A rocket, continuously accelerated by its exhaust, can escape without ever reaching escape speed, since it continues to add kinetic energy from its engines. It can achieve escape at any speed, given sufficient propellant to provide new acceleration to the rocket to counter gravity's deceleration and thus maintain its speed.

More generally, escape velocity is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero; [nb 1] an object which has achieved escape velocity is neither on the surface, nor in a closed orbit (of any radius). With escape velocity in a direction pointing away from the ground of a massive body, the object will move away from the body, slowing forever and approaching, but never reaching, zero speed. Once escape velocity is achieved, no further impulse need be applied for it to continue in its escape. In other words, if given escape velocity, the object will move away from the other body, continually slowing, and will asymptotically approach zero speed as the object's distance approaches infinity, never to come back. [1] Speeds higher than escape velocity retain a positive speed at infinite distance. Note that the minimum escape velocity assumes that there is no friction (e.g., atmospheric drag), which would increase the required instantaneous velocity to escape the gravitational influence, and that there will be no future acceleration or extraneous deceleration (for example from thrust or from gravity of other bodies), which would change the required instantaneous velocity.

Escape speed at a distance d from the center of a spherically symmetric primary body (such as a star or a planet) with mass M is given by the formula [2]

where G is the universal gravitational constant (G ≈ 6.67×10−11 m3·kg−1·s−2). [nb 2] The escape speed is independent of the mass of the escaping object. For example, the escape speed from Earth's surface is about 11.186 km/s (40,270 km/h; 25,020 mph; 36,700 ft/s). [3]

When given an initial speed greater than the escape speed the object will asymptotically approach the hyperbolic excess speed satisfying the equation: [4]

In these equations atmospheric friction (air drag) is not taken into account.

Overview

Luna 1, launched in 1959, was the first man-made object to attain escape velocity from Earth (see below table). RIAN archive 510848 Interplanetary station Luna 1 - blacked.jpg
Luna 1, launched in 1959, was the first man-made object to attain escape velocity from Earth (see below table).

The existence of escape velocity is a consequence of conservation of energy and an energy field of finite depth. For an object with a given total energy, which is moving subject to conservative forces (such as a static gravity field) it is only possible for the object to reach combinations of locations and speeds which have that total energy; and places which have a higher potential energy than this cannot be reached at all. By adding speed (kinetic energy) to the object it expands the possible locations that can be reached, until, with enough energy, they become infinite.

For a given gravitational potential energy at a given position, the escape velocity is the minimum speed an object without propulsion needs to be able to "escape" from the gravity (i.e. so that gravity will never manage to pull it back). Escape velocity is actually a speed (not a velocity) because it does not specify a direction: no matter what the direction of travel is, the object can escape the gravitational field (provided its path does not intersect the planet).

An elegant way to derive the formula for escape velocity is to use the principle of conservation of energy (for another way, based on work, see below). For the sake of simplicity, unless stated otherwise, we assume that an object will escape the gravitational field of a uniform spherical planet by moving away from it and that the only significant force acting on the moving object is the planet's gravity. Imagine that a spaceship of mass m is initially at a distance r from the center of mass of the planet, whose mass is M, and its initial speed is equal to its escape velocity, . At its final state, it will be an infinite distance away from the planet, and its speed will be negligibly small. Kinetic energy K and gravitational potential energy Ug are the only types of energy that we will deal with (we will ignore the drag of the atmosphere), so by the conservation of energy,

We can set Kfinal = 0 because final velocity is arbitrarily small, and Ugfinal = 0 because final distance is infinity, so

where μ is the standard gravitational parameter.

The same result is obtained by a relativistic calculation, in which case the variable r represents the radial coordinate or reduced circumference of the Schwarzschild metric. [6] [7]

Defined a little more formally, "escape velocity" is the initial speed required to go from an initial point in a gravitational potential field to infinity and end at infinity with a residual speed of zero, without any additional acceleration. [8] All speeds and velocities are measured with respect to the field. Additionally, the escape velocity at a point in space is equal to the speed that an object would have if it started at rest from an infinite distance and was pulled by gravity to that point.

In common usage, the initial point is on the surface of a planet or moon. On the surface of the Earth, the escape velocity is about 11.2 km/s, which is approximately 33 times the speed of sound (Mach 33) and several times the muzzle velocity of a rifle bullet (up to 1.7 km/s). However, at 9,000 km altitude in "space", it is slightly less than 7.1 km/s. Note that this escape velocity is relative to a non-rotating frame of reference, not relative to the moving surface of the planet or moon (see below).

The escape velocity is independent of the mass of the escaping object. It does not matter if the mass is 1 kg or 1,000 kg; what differs is the amount of energy required. For an object of mass the energy required to escape the Earth's gravitational field is GMm / r, a function of the object's mass (where r is radius of the Earth, nominally 6,371 kilometres (3,959 mi), G is the gravitational constant, and M is the mass of the Earth, M = 5.9736 × 1024 kg). A related quantity is the specific orbital energy which is essentially the sum of the kinetic and potential energy divided by the mass. An object has reached escape velocity when the specific orbital energy is greater than or equal to zero.

Scenarios

From the surface of a body

An alternative expression for the escape velocity particularly useful at the surface on the body is:

where r is the distance between the center of the body and the point at which escape velocity is being calculated and g is the gravitational acceleration at that distance (i.e., the surface gravity). [9]

For a body with a spherically-symmetric distribution of mass, the escape velocity from the surface is proportional to the radius assuming constant density, and proportional to the square root of the average density ρ.

where

Note that this escape velocity is relative to a non-rotating frame of reference, not relative to the moving surface of the planet or moon, as we now explain.

From a rotating body

The escape velocity relative to the surface of a rotating body depends on direction in which the escaping body travels. For example, as the Earth's rotational velocity is 465 m/s at the equator, a rocket launched tangentially from the Earth's equator to the east requires an initial velocity of about 10.735 km/s relative to the moving surface at the point of launch to escape whereas a rocket launched tangentially from the Earth's equator to the west requires an initial velocity of about 11.665 km/s relative to that moving surface. The surface velocity decreases with the cosine of the geographic latitude, so space launch facilities are often located as close to the equator as feasible, e.g. the American Cape Canaveral (latitude 28°28 N) and the French Guiana Space Centre (latitude 5°14 N).

Practical considerations

In most situations it is impractical to achieve escape velocity almost instantly, because of the acceleration implied, and also because if there is an atmosphere, the hypersonic speeds involved (on Earth a speed of 11.2 km/s, or 40,320 km/h) would cause most objects to burn up due to aerodynamic heating or be torn apart by atmospheric drag. For an actual escape orbit, a spacecraft will accelerate steadily out of the atmosphere until it reaches the escape velocity appropriate for its altitude (which will be less than on the surface). In many cases, the spacecraft may be first placed in a parking orbit (e.g. a low Earth orbit at 160–2,000 km) and then accelerated to the escape velocity at that altitude, which will be slightly lower (about 11.0 km/s at a low Earth orbit of 200 km). The required additional change in speed, however, is far less because the spacecraft already has a significant orbital speed (in low Earth orbit speed is approximately 7.8 km/s, or 28,080 km/h).

From an orbiting body

The escape velocity at a given height is times the speed in a circular orbit at the same height, (compare this with the velocity equation in circular orbit). This corresponds to the fact that the potential energy with respect to infinity of an object in such an orbit is minus two times its kinetic energy, while to escape the sum of potential and kinetic energy needs to be at least zero. The velocity corresponding to the circular orbit is sometimes called the first cosmic velocity, whereas in this context the escape velocity is referred to as the second cosmic velocity. [10]

For a body in an elliptical orbit wishing to accelerate to an escape orbit the required speed will vary, and will be greatest at periapsis when the body is closest to the central body. However, the orbital speed of the body will also be at its highest at this point, and the change in velocity required will be at its lowest, as explained by the Oberth effect.

Barycentric escape velocity

Technically escape velocity can either be measured as a relative to the other, central body or relative to center of mass or barycenter of the system of bodies. Thus for systems of two bodies, the term escape velocity can be ambiguous, but it is usually intended to mean the barycentric escape velocity of the less massive body. In gravitational fields, escape velocity refers to the escape velocity of zero mass test particles relative to the barycenter of the masses generating the field. In most situations involving spacecraft the difference is negligible. For a mass equal to a Saturn V rocket, the escape velocity relative to the launch pad is 253.5 am/s (8 nanometers per year) faster than the escape velocity relative to the mutual center of mass.[ citation needed ]

Height of lower-velocity trajectories

Ignoring all factors other than the gravitational force between the body and the object, an object projected vertically at speed from the surface of a spherical body with escape velocity and radius will attain a maximum height satisfying the equation [11]

which, solving for h results in

where is the ratio of the original speed to the escape velocity

Unlike escape velocity, the direction (vertically up) is important to achieve maximum height.

Trajectory

If an object attains exactly escape velocity, but is not directed straight away from the planet, then it will follow a curved path or trajectory. Although this trajectory does not form a closed shape, it can be referred to as an orbit. Assuming that gravity is the only significant force in the system, this object's speed at any point in the trajectory will be equal to the escape velocity at that point due to the conservation of energy, its total energy must always be 0, which implies that it always has escape velocity; see the derivation above. The shape of the trajectory will be a parabola whose focus is located at the center of mass of the planet. An actual escape requires a course with a trajectory that does not intersect with the planet, or its atmosphere, since this would cause the object to crash. When moving away from the source, this path is called an escape orbit. Escape orbits are known as C3 = 0 orbits. C3 is the characteristic energy, = −GM/2a, where a is the semi-major axis, which is infinite for parabolic trajectories.

If the body has a velocity greater than escape velocity then its path will form a hyperbolic trajectory and it will have an excess hyperbolic velocity, equivalent to the extra energy the body has. A relatively small extra delta-v above that needed to accelerate to the escape speed can result in a relatively large speed at infinity. Some orbital manoeuvres make use of this fact. For example, at a place where escape speed is 11.2 km/s, the addition of 0.4 km/s yields a hyperbolic excess speed of 3.02 km/s:

If a body in circular orbit (or at the periapsis of an elliptical orbit) accelerates along its direction of travel to escape velocity, the point of acceleration will form the periapsis of the escape trajectory. The eventual direction of travel will be at 90 degrees to the direction at the point of acceleration. If the body accelerates to beyond escape velocity the eventual direction of travel will be at a smaller angle, and indicated by one of the asymptotes of the hyperbolic trajectory it is now taking. This means the timing of the acceleration is critical if the intention is to escape in a particular direction.

If the speed at periapsis is v, then the eccentricity of the trajectory is given by:

This is valid for elliptical, parabolic, and hyperbolic trajectories. If the trajectory is hyperbolic or parabolic, it will asymptotically approach an angle from the direction at periapsis, with

The speed will asymptotically approach

List of escape velocities

In this table, the left-hand half gives the escape velocity from the visible surface (which may be gaseous as with Jupiter for example), relative to the centre of the planet or moon (that is, not relative to its moving surface). In the right-hand half, Ve refers to the speed relative to the central body (for example the sun), whereas Vte is the speed (at the visible surface of the smaller body) relative to the smaller body (planet or moon).

LocationRelative toVe (km/s) [12] LocationRelative toVe (km/s) [12] System escape, Vte (km/s)
On the Sun The Sun's gravity617.5
On Mercury Mercury's gravity4.25At MercuryThe Sun's gravity~ 67.7~ 20.3
On Venus Venus's gravity10.36At VenusThe Sun's gravity49.517.8
On Earth Earth's gravity11.186At EarthThe Sun's gravity42.116.6
On the Moon The Moon's gravity2.38At the MoonThe Earth's gravity1.42.42
On Mars Mars' gravity5.03At MarsThe Sun's gravity34.111.2
On Ceres Ceres's gravity0.51At CeresThe Sun's gravity25.37.4
On Jupiter Jupiter's gravity60.20At JupiterThe Sun's gravity18.560.4
On Io Io's gravity2.558At IoJupiter's gravity24.57.6
On Europa Europa's gravity2.025At EuropaJupiter's gravity19.46.0
On Ganymede Ganymede's gravity2.741At GanymedeJupiter's gravity15.45.3
On Callisto Callisto's gravity2.440At CallistoJupiter's gravity11.64.2
On Saturn Saturn's gravity36.09At SaturnThe Sun's gravity13.636.3
On Titan Titan's gravity2.639At TitanSaturn's gravity7.83.5
On Uranus Uranus' gravity21.38At UranusThe Sun's gravity9.621.5
On Neptune Neptune's gravity23.56At NeptuneThe Sun's gravity7.723.7
On Triton Triton's gravity1.455At TritonNeptune's gravity6.22.33
On Pluto Pluto's gravity1.23At PlutoThe Sun's gravity~ 6.6~ 2.3
At Solar System galactic radiusThe Milky Way's gravity492594 [13] [14]
On the event horizon A black hole's gravity299,792.458 (speed of light)

The last two columns will depend precisely where in orbit escape velocity is reached, as the orbits are not exactly circular (particularly Mercury and Pluto).

Deriving escape velocity using calculus

Let G be the gravitational constant and let M be the mass of the earth (or other gravitating body) and m be the mass of the escaping body or projectile. At a distance r from the centre of gravitation the body feels an attractive force

The work needed to move the body over a small distance dr against this force is therefore given by

The total work needed to move the body from the surface r0 of the gravitating body to infinity is then [15]

In order to do this work to reach infinity, the body's minimal kinetic energy at departure must match this work, so the escape velocity v0 satisfies

which results in

See also

Notes

  1. The gravitational potential energy is negative since gravity is an attractive force and the potential energy has been defined for this purpose to be zero at infinite distance from the centre of gravity.
  2. The value GM is called the standard gravitational parameter, or μ, and is often known more accurately than either G or M separately.

Related Research Articles

Orbit Curved path of an object around a point

In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. In an atom, electrons follow similar curved paths, or orbits, around a nucleus. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

Free fall Motion of a body subject only to gravity

In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

Orbital period Time an astronomical object takes to complete one orbit around another object

The orbital period is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars.

Gravity assist Space navigation technique

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 astronomical object to alter the path and speed of a spacecraft, typically to save propellant and reduce expense.

Hohmann transfer orbit Elliptical orbit used to transfer between two orbits of different altitudes, in the same plane

In orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits of different radii around a central body in the same plane. The Hohmann transfer often uses the lowest possible amount of propellant in traveling between these orbits, but bi-elliptic transfers can use less in some cases.

Orbital mechanics Field of classical mechanics concerned with the motion of spacecraft

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

Orbital speed Speed at which a body orbits around the barycenter of a system

In gravitationally bound systems, the orbital speed of an astronomical body or object is the speed at which it orbits around either the barycenter or, if one object is much more massive than the other bodies in the system, its speed relative to the center of mass of the most massive body.

Hill sphere Region in which an astronomical body dominates the attraction of satellites

The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.

Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).

Hyperbolic trajectory

In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In astrodynamics, the characteristic energy is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass.

Elliptic orbit Kepler orbit with an eccentricity of less than one

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

Circular orbit Orbit with a fixed distance from the barycenter

A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.

Specific orbital energy

In the gravitational two-body problem, the specific orbital energy of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy, divided by the reduced mass. According to the orbital energy conservation equation, it does not vary with time:

In astrodynamics an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or the focus.

Flight dynamics (spacecraft) Application of mechanical dynamics to model the flight of space vehicles

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

A gravity train is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the interior of the sphere.

A set of equations describe the resultant trajectories when objects move owing to a constant gravitational force under normal Earth-bound conditions. For example, Newton's law of universal gravitation simplifies to F = mg, where m is the mass of the body. This assumption is reasonable for objects falling to earth over the relatively short vertical distances of our everyday experience, but is untrue over larger distances, such as spacecraft trajectories.

Semi-major and semi-minor axes Term in geometry; longest and shortest semidiameters of an ellipse

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

Radial trajectory

In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

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