In physics, escape velocity is the speed at which the kinetic energy plus the gravitational
potential energy of an object is zero. It is the speed needed to "break free" from the
gravitational attraction of a massive body, without further propulsion, i.e., without
spending more fuel. For a spherically symmetric body, the escape
velocity at a given distance is calculated by the formula
where G is the universal gravitational constant, M the mass of the planet, star or other body,
and r the distance from the center of gravity. In this equation atmospheric friction is not
taken into account. A rocket moving out of a gravity well does not actually need to attain
escape velocity to do so, but could achieve the same result at any speed with a suitable
mode of propulsion and sufficient fuel. Escape velocity only applies to ballistic trajectories.
The term escape velocity is actually a misnomer, and it is often more accurately referred to
as escape speed since the necessary speed is a scalar quantity which is independent
of direction.
Overview
A barycentric velocity is a velocity of one body relative to the center of mass of a system
of bodies. A relative velocity is the velocity of one body with respect to another. Relative
escape velocity is defined only in systems with two bodies. For systems of two bodies
the term "escape velocity" is 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. The existence of escape velocity is a consequence
of conservation of energy. For an object with a given total energy, which is moving subject
to conservative forces 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 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. For the sake of simplicity, unless stated
otherwise, we will assume that an object is attempting to escape from a uniform spherical
planet by moving directly away from it and that the only significant force acting on
the moving object is the planet's gravity. Escape velocity is actually a speed because
it does not specify a direction: no matter what the direction of travel is, the object
can escape the gravitational field. 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 planet or moon. On the surface
of the Earth, the escape velocity is about 11.2 kilometers per second, which is approximately
33 times the speed of sound and several times the muzzle velocity 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 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 and the French Guiana Space Centre.
The barycentric 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. For a mass equal to a Saturn V rocket, the
escape velocity relative to the launch pad is 253.5 am/s faster than the escape velocity
relative to the mutual center of mass. When the mass reaches the Andromeda galaxy the
earth will have recoiled 500 m. 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, 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. The shape of the orbit will be a parabola whose focus is located at the
center of mass of the planet. An actual escape requires a course with an orbit 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/a, where a is the semi-major
axis, which is infinite for parabolic orbits. When there are many gravitating bodies, such
as in the solar system, a rocket that travels at escape velocity from one body, say Earth,
will not travel to an infinite distance because it needs an even higher speed to escape the
Sun's gravity. Near the Earth, the rocket's orbit will appear parabolic, but will become
an ellipse around the Sun, except when it is perturbed by the Earth, whose orbit it
must still intersect, and other bodies. List of escape velocities
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 hypersonic
regime for most practical propulsion systems and 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 is first placed in low Earth orbit and then accelerated to the escape velocity
at that altitude, which is a little less — about 10.9 km/s. The required change in speed,
however, is 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 barycentric escape velocity, G is the gravitational constant, M is the
mass of the body being escaped from, r is the distance between the center of the body
and the point at which escape velocity is being calculated, g is the gravitational acceleration
at that distance, and μ is the standard gravitational parameter.
The escape velocity at a given height is times the speed in a circular orbit at the same
height,. 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
For a body with a spherically-symmetric distribution of mass, the barycentric escape velocity from
the surface is approximately 2.364×10−5 m1.5kg−0.5s−1 times the radius r times
the square root of the average density ρ, or:
Deriving escape velocity using calculus Let G be the gravitational constant and let
M be the mass of the earth 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
This is the minimal required kinetic energy to be able to reach infinity, so the escape
velocity v0 satisfies
which results in
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 it can be shown that this gives an escape
velocity which is equal to the square root of the sum of the squares of the individual
escape velocities due to each source. For example, at the Earth's surface the escape
velocity for the combination Earth and Sun would be .
See also
Orbital speed Characteristic energy
Delta-v budget – speed needed to perform manoeuvres.
Gravitational slingshot – a technique for changing trajectory.
Gravity well Two-body problem
Black hole – an object with an escape velocity greater than the speed of light
Oberth effect – burning fuel deep in a gravity field gives higher change in velocity.
Newton's cannonball List of artificial objects escaping from the
Solar System List of artificial objects in heliocentric
orbit Notes
^ 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. ^ The value GM is called the standard gravitational
parameter, or μ, and is often known more accurately than either G or M separately.
References Roger R. Bate, Donald D. Mueller, and Jerry
E. White. Fundamentals of astrodynamics. New York: Dover Publications. ISBN 0-486-60061-0.
External links Web-based numerical escape velocity calculator