In
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
,
escape velocity is the
speedSpeed is the rate of motion, or equivalently the rate of change of distance.Speed is a scalar quantity with dimensions length/time; the equivalent vector quantity to speed is velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of...
where the
kinetic energyThe kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its...
of an object is equal to the magnitude of its gravitational potential energy, as calculated by the equation,
It is commonly described as the speed needed to "break free" from a gravitational field (without any additional impulse) and is theoretical, totally neglecting atmospheric friction. The term
escape velocity can be considered a
misnomerA misnomer is a term which suggests an interpretation that is known to be untrue. Such incorrect terms sometimes derived their names because of the form, action, or origin of the subject becoming named popularly or widely referenced—long before their true natures were known.- Sources of misnomers...
because it is actually a speed rather than a
velocityIn physics, velocity is the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI system, it is measured in meters per second: or ms-1. The scalar absolute value of velocity is speed...
, i.e. it specifies how fast the object must move but the direction of movement is irrelevant, unless "downward." In more technical terms, escape velocity is a
scalarIn physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . -Physical quantity:...
(and not a vector).
Escape velocity gives a minimum
delta-v budgetDelta-v budget is an astrogation term used in astrodynamics and aerospace industry for velocity change requirements for the various propulsive tasks and orbital maneuvers over phases of a space mission....
for rockets when no benefit can be obtained from the speeds of other bodies for a particular mission; but it neglects losses such as air drag and
gravity 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...
. However in some cases it can be improved upon, for example, by use of
gravitational slingshotIn orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist 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 fuel, time, and expense. Gravity assist...
s.
Overview
The phenomenon of escape velocity is a consequence of
conservation of energyThe law of conservation of energy states that the total amount of energy in a closed system remains constant. A consequence of this law is that energy cannot be created nor destroyed...
. For an object with a given total energy, which is moving subject to
conservative forceA conservative force is defined as a force with the following property: when a particle moves in any closed loop, the force acting along the path multiplied by the distance travelled always sums to zero....
s (such as a static gravity field) it is only possible for the object to reach combinations of places and speeds which have that total energy; and places which have a higher potential energy than this cannot be reached at all.
For a given
gravitational potentialIn celestial mechanics, the gravitational potential belonging to an individual massive object, such as the earth or the sun, is a scalar field generated by that object...
energy at a given position, the
escape velocity is the minimum
speedSpeed is the rate of motion, or equivalently the rate of change of distance.Speed is a scalar quantity with dimensions length/time; the equivalent vector quantity to speed is velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of...
an object without
propulsionSpacecraft propulsion is any method used to accelerate spacecraft and artificial satellites. There are many different methods. Each method has drawbacks and advantages, and spacecraft propulsion is an active area of research. However, most spacecraft today are propelled by exhausting a gas from the...
needs to have sufficient energy to be able to "escape" from the gravity, i.e. so that gravity will never manage to pull it back. For the sake of simplicity, unless stated otherwise, we will assume that the scenario we are dealing with is that an object is attempting to escape from a uniform spherical planet by moving straight up (along a radial line away from the center of the planet), and that the
only significant force acting on the moving object is the planet's gravity.
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 (though its path may intersect the planet). The simplest way of deriving the formula for escape velocity is to use conservation of energy. Imagine that a spaceship of mass
m is at a distance
r from the center of mass of the planet, whose mass is
M. 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 and assumed to be 0. Kinetic energy
K and gravitational potential energy
Ug are the only types of energy that we will deal with, so by the conservation of
'energy',
Kƒ = 0 because final velocity is zero, and
Ugƒ = 0 because its final distance is infinity, so
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 with a residual velocity of zero, with all speeds and velocities 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
planetA 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
moonA natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called the primary. Technically, the term natural satellite could refer to a planet orbiting a star, or a dwarf galaxy orbiting a major galaxy, but it is normally synonymous with moon and used to identify...
. On the surface of the Earth, the escape velocity is about 11.2 kilometers per second (~6.96 mi/s), which is approximately 34 times the speed of sound (mach 34) and at least 10 times the speed of a rifle bullet. However, at 9,000 km altitude in "space", it is slightly less than 7.1 km/s.
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 Earth 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 Earth. The surface velocity decreases with the
cosineIn mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
of the geographic latitude, so space launch facilities are often located as close to the equator as feasible, e.g. the American
Cape CanaveralThe Cape Canaveral Air Force Station is an installation of the Air Force Space Command's 45th Space Wing , headquartered at nearby Patrick Air Force Base. Located on Cape Canaveral in the State of Florida, CCAFS is the primary Launch Head of the Eastern Range...
(latitude 28°28' N) and the French Guiana Space Centre (latitude 5°14' N).
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, escape velocity from the same point in the same gravitational field is always the same. What differs is the amount of energy needed to accelerate the mass to achieve escape velocity: the energy needed for an object of mass to escape the Earth's gravitational field is
GMm / r, a function of the object's mass (where
r is the radius of the Earth,
G is the
gravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
, and
M is the mass of the Earth). More massive objects require more energy to reach escape velocity. All of this, of course, assumes we are neglecting air resistance.
Misconception
Planetary or lunar escape velocity is sometimes misunderstood to be the speed a powered vehicle (such as a rocket)
must reach to leave orbit; however, this is not the case, as the quoted number is typically the
surface escape velocity, and vehicles need never achieve that speed. This surface escape velocity is the speed required for an object to leave the planet if the object is simply projected from the surface of the planet and then left without any more kinetic energy input: in practice the vehicle's propulsion system will continue to provide energy after it has left the surface.
In fact a vehicle can leave the Earth's gravity at any speed. At higher altitudes, the local escape velocity is lower. But at the instant the propulsion stops, the vehicle can only escape if its speed is greater than or equal to the local escape velocity at
that position. At sufficiently high altitudes this speed can approach 0.
Orbit
If an object attains escape velocity, but is not directed straight away from the planet, then it will follow a curved path. Although this path does not form a closed shape, in astrodynamic terminology it is still considered an orbit. Assuming that gravity is the only significant force in the system, this object's speed at any point in the orbit 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 orbit will be a
parabolaIn 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...
whose focus is located at the center of mass of the planet. An actual escape requires of course that the orbit not intersect the planet nor its atmosphere, since this would cause the object to crash. When moving away from the source, this path is called an
escape orbitAn escape orbit is a high-energy parabolic orbit around the central body. A body in this orbit has at each position the escape velocity with respect to this central body, for this position...
; when moving closer to the source, a
capture orbitA capture orbit is a reverse escape orbit. It is a parabolic orbit with as special case a straight line in the direction of the center of the central body. If it intersects the central body or its atmosphere the object will crash into the central body or there will be atmospheric re-entry...
. Both are known as
C3 = 0 orbits (where
C3 = -
μ/a, and
a is the
semi-major axisIn geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae.- Ellipse :The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
).
In reality there are many gravitating bodies in space, so that, for instance, a rocket that travels at escape velocity from Earth will not escape to an infinite distance away because it needs an even higher speed to escape the Sun's gravity. In other words, near the Earth, the rocket's orbit will appear parabolic, but eventually its orbit will become an ellipse around the Sun, except when it is perturbed by the Earth whose orbit it must still intersect.
However, it is also easily possible to calculate the escape velocity that will escape from all gravitating bodies.
List of escape velocities
| Location |
with respect to |
Ve |
|
Location |
with respect to |
Ve |
| on the Sun The Sun is the star at the center of the Solar System. The Earth and other matter orbit the Sun, which by itself accounts for about 99.86% of the Solar System's mass....
, |
the Sun's gravity: |
617.5 km/s |
|
|
|
on MercuryFor the liquid metallic element, see Mercury .Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three...
, |
Mercury's gravity: |
4.3 km/s |
|
at Mercury, |
the Sun's gravity: |
67.7 km/s |
on VenusVenus is the second-closest planet to the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6...
, |
Venus' gravity: |
10.3 km/s |
|
at Venus, |
the Sun's gravity: |
49.5 km/s |
on EarthEarth is the third planet from the Sun. It is the fifth largest of the eight planets in the solar system, and the largest of the terrestrial planets in the Solar System in terms of diameter, mass and density...
, |
the Earth's gravity: |
11.2 km/s |
|
at the Earth/Moon, |
the Sun's gravity: |
42.1 km/s |
on the MoonThe Moon is Earth's only natural satellite and the fifth largest satellite in the Solar System. The average centre-to-centre distance from the Earth to the Moon is , about thirty times the diameter of the Earth. The common centre of mass of the system is located at about —a quarter the Earth's...
, |
the Moon's gravity: |
2.4 km/s |
|
at the Moon, |
the Earth's gravity: |
1.4 km/s |
on MarsMars is the fourth planet from the Sun in the Solar System. The planet is named after Mars, the Roman god of war. It is also referred to as the "Red Planet" because of its reddish appearance, due to iron oxide prevalent on its surface....
, |
Mars' gravity: |
5.0 km/s |
|
at Mars, |
the Sun's gravity: |
34.1 km/s |
on JupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with a mass slightly less than one-thousandth that of the Sun but is two and a half times the mass of all of the other planets in our Solar System combined. Jupiter is classified as a gas...
, |
Jupiter's gravity: |
59.5 km/s |
|
at Jupiter, |
the Sun's gravity: |
18.5 km/s |
| on Saturn Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn, along with Jupiter, Uranus and Neptune, is classified as a gas giant...
, |
Saturn's gravity: |
35.6 km/s |
|
at Saturn, |
the Sun's gravity: |
13.6 km/s |
on UranusUranus is the seventh planet from the Sun, and the third-largest and fourth most massive planet in the Solar System. It is named after the ancient Greek deity of the sky Uranus the father of Kronos and grandfather of Zeus...
, |
Uranus' gravity: |
21.2 km/s |
|
at Uranus, |
the Sun's gravity: |
9.6 km/s |
on NeptuneNeptune is the eighth planet from the Sun in our Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third-largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 Earth masses and...
, |
Neptune's gravity: |
23.6 km/s |
|
at Neptune, |
the Sun's gravity: |
7.7 km/s |
in the solar systemThe Solar System consists of the Sun and those celestial objects bound to it by gravity, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago...
, |
the Milky WayThe Milky Way, or simply the Galaxy, is the galaxy in which the Solar System is located. It is a barred spiral galaxy that is part of the Local Group of galaxies...
's gravity: |
~1,000 km/s |
|
|
|
on the event horizonIn general relativity, an event horizon is a boundary in spacetime, most often an area surrounding a black hole, beyond which events cannot affect an outside observer...
, |
the black holeIn general relativity, a black hole is a region of space in which the gravitational field is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come...
's gravity: |
superluminal |
|
|
|
Because of the atmosphere it is not useful and hardly possible to give an object near the surface of the Earth a speed of 11.2 km/s, as these speeds are too far in the
hypersonicIn aerodynamics, hypersonic speeds are those that are highly supersonic. Since the 1970s, the term has generally been assumed to refer to speeds of Mach 5 and above...
regime for most practical propulsion systems and would cause most objects to burn up due to atmospheric compression or be torn apart by atmospheric friction. For an actual escape orbit a spacecraft is first placed in
low Earth orbitA low Earth orbit is generally defined as an orbit within the locus extending from the Earth’s surface up to an altitude of 2,000 km...
and then accelerated to the escape velocity at that altitude, which is a little less — about 10.9 km/s. The required acceleration, however, is generally far less because from a low Earth orbit the spacecraft already has a speed of approximately 8 km/s.
Calculating an escape velocity
To expand upon the derivation given in the Overview,
where is the escape velocity,
G is the
gravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
,
M is the
massIn physics, mass commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: inertial mass, active gravitational mass and passive gravitational mass...
of the body being escaped from,
r is the
distanceDistance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical...
between the center of the body and the point at which escape velocity is being calculated,
g is the
gravitational accelerationIn physics, gravitational acceleration is often described as the acceleration of an object caused by the force of gravity from another object. Any object will accelerate in a gravitational field at the same rate, regardless of the mass of the object...
at that distance, and μ is the
standard gravitational parameterIn astrodynamics, the standard gravitational parameter of a celestial body is the product of the gravitational constant and the mass :The units of the standard gravitational parameter are km3s-2...
.
The escape velocity at a given height is times the speed in a circular orbit at the same height (compare this with equation (14) in
circular motionIn physics, circular motion is rotation along a circle: a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation, or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular...
). 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.
For a body with a spherically-symmetric distribution of mass, the escape velocity from the surface (in m/s) is approximately 2.364×10
−5 m
1.5kg
−0.5s
−1 times the radius
r (in meters) times the square root of the average density ρ (in kg/m³), or:
Deriving escape velocity using calculus
These derivations use
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
,
Newton's laws of motionNewton's laws of motion are three physical laws that form the basis for classical mechanics. They are:# In the absence of force, a body either is at rest or moves in a straight line with constant speed....
and
Newton's law of universal gravitationNewton's law of universal gravitation states that every object in this universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of distance between their centres. This is a general physical law derived...
.
Derivation using only g and r
the Earth's escape speed can be derived from "
gStandard gravity, usually denoted by g0 or gn, is the nominal acceleration due to gravity at the Earth's surface at sea level, defined to be precisely 9.80665 m/s2 . This value was established by the 3rd CGPM...
," the acceleration due to gravity at the Earth's surface. It is not necessary to know the
gravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
G or the mass
M of the Earth. Let
- r = the Earth's radius, and
- g = the acceleration of gravity at the Earth's surface.
Above the Earth's surface, the acceleration of gravity is governed by Newton's
inverse-squareIn physics, an inverse-square law is any physical law stating that some physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity.- Justification :...
law of universal gravitationNewton's law of universal gravitation states that every object in this universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of distance between their centres. This is a general physical law derived...
. Accordingly, the acceleration of gravity at height
s above the center of the Earth (where
s >
r ) is
g (
r /
s)
2.
The weight of an object of mass
m at the surface is
g m, and its weight at height
s above the center of the Earth is
gm (
r /
s)
2. Consequently the energy needed to lift an object of mass
m from height
s above the Earth's center to height
s +
ds (where
ds is an
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
increment of
s) is
gm (
r /
s)
2 ds. Since this decreases sufficiently fast as
s increases, the total energy needed to lift the object to infinite height does not diverge to infinity, but converges to a finite amount. That amount is the integral of the expression above:
That is how much
kinetic energy the object of mass
m needs in order to escape. The kinetic energy of an object of mass
m moving at speed
v is (1/2)
mv2. Thus we need
The factor
m cancels out, and solving for
v we get
If we take the radius of the Earth to be
r = 6400 kilometers and the acceleration of gravity at the surface to be
g = 9.8 m/s
2, we get
This is just a bit over 11 kilometers per second, or a bit under 7 miles per second, as
Isaac NewtonSir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is perceived and considered by a substantial number of scholars and the general public as one of the most influential men in history...
calculated.
Derivation using G and M
Let
G be the
gravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
and let
M be the mass of the earth or other body to be escaped.
By applying the
chain ruleIn calculus, the chain rule is a formula for the derivative of the composite of two functions.In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of y with respect to x can be computed as the rate of change of...
, we get:
Because
Since we want escape velocity and
v0 is the escape velocity and
r0 is the radius of the planet. Note that the above derivation relies on the equivalence of inertial mass and gravitational mass.
The derivations are consistent
The gravitational acceleration can be obtained from the gravitational constant
G and the mass of Earth
M:
where
r is the radius of Earth. Thus
so the two derivations given above are consistent.
Multiple sources
The escape velocity from a position in a field with multiple sources at rest with respect to each other is derived from the total potential energy per kg at that position, relative to infinity. The potential energies for all sources can simply be added. For the escape velocity this results in the square root of the sum of the squares of the escape velocities of all sources separately.
For example, at the Earth's surface the escape velocity for the combination Earth and Sun would be , if the Earth were stationary with respect to the sun. Since it is not, the minimum effective escape velocity is reduced by the orbital velocity, which, in the circular orbit approximation, means it would be multiplied by a factor of . Therefore the minimum actual combined Earth-Sun escape velocity from the Earth's surface is . As a result, to leave the solar system requires a speed of 16.7 km/s from the Earth's surface; in the direction of the Earth's orbital motion.
Dynamic situations
The escape velocity calculation assumes that none of the sources of gravitational potential are moving. In the case of the Earth-Moon system for example the Moon is in orbit around the Earth, and it turns out that an object that passes behind the moon can gain speed from the encounter (a gravitational assist).
Therefore there are certain launch angles and
launch windowLaunch window is a term used in spaceflight to describe a time period in which a particular rocket must be launched. If the rocket does not launch within the "window", it has to wait for the next window....
s that will give the assist and this will give an object extra speed in-flight and still permit it to escape Earth's gravity even though it was launched with less than escape velocity.
Gravity well
In the hypothetical case of uniform density, the velocity that an object would achieve when dropped in a hypothetical vacuum hole from the surface of the Earth to the center of the Earth is the escape velocity divided by , i.e. the speed in a circular orbit at a low height. Correspondingly, the escape velocity from the center of the Earth would be times that from the surface.
A refined calculation would take into account the fact that the Earth's mass is not uniformly distributed as the center is approached. This gives higher speeds.
See also
- Gravitational potential energy
- Delta-v budget
Delta-v budget is an astrogation term used in astrodynamics and aerospace industry for velocity change requirements for the various propulsive tasks and orbital maneuvers over phases of a space mission....
– speed needed to perform manoeuvres.
- Gravitational slingshot
In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist 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 fuel, time, and expense. Gravity assist...
– three-body technique for gaining energy
- Gravity well
In physics, a gravity well is the gravitational potential field around a massive body . Physical models of gravity wells are sometimes used to illustrate orbital mechanics...
- Two-body problem
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic...
- Black hole
In general relativity, a black hole is a region of space in which the gravitational field is so powerful that nothing, not even light, can escape. The black hole has a one-way surface, called an event horizon, into which objects can fall, but out of which nothing can come...
– An object with an escape velocity greater than the speed of light
- Oberth effect
The Oberth effect or powered flyby is a feature of astronautics where use of a rocket engine close to a gravitational body can give much more kinetic energy and a much bigger change in final speed than the same burn executed further from the body for the same initial orbit...
– burning fuel deep in a gravity field gives higher velocity
External links