The Binet equation, derived by
Jacques Philippe Marie Binet
Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical founda ...
, provides the form of a
central force
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.
: \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat
where \vec F is the force, F is a vecto ...
given the shape of the
orbital motion
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 p ...
in plane
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
. The equation can also be used to derive the shape of the orbit for a given force law, but this usually involves the solution to a second order
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
. A unique solution is impossible in the case of
circular motion
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of ro ...
about the center of force.
Equation
The shape of an orbit is often conveniently described in terms of relative distance
as a function of angle
. For the Binet equation, the orbital shape is instead more concisely described by the reciprocal
as a function of
. Define the
specific angular momentum
In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative positi ...
as
where
is the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
and
is the mass. The Binet equation, derived in the next section, gives the force in terms of the function
:
:
Derivation
Newton's Second Law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
for a purely central force is
:
The
conservation of angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system ...
requires that
:
Derivatives of
with respect to time may be rewritten as derivatives of
with respect to angle:
:
Combining all of the above, we arrive at
:
The general solution is
where
is the initial coordinate of the particle.
Examples
Kepler problem
Classical
The traditional
Kepler problem
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
of calculating the orbit of an
inverse square law
In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understoo ...
may be read off from the Binet equation as the solution to the differential equation
:
:
If the angle
is measured from the
periapsis
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary (astronomy), primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two ...
, then the general solution for the orbit expressed in (reciprocal) polar coordinates is
:
The above polar equation describes
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s, with
the
semi-latus rectum
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
(equal to
) and
the
orbital eccentricity
In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values betwee ...
.
Relativistic
The relativistic equation derived for
Schwarzschild coordinates In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordi ...
is
:
where
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and
is the
Schwarzschild radius
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic ...
. And for
Reissner–Nordström metric
In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass ''M''. T ...
we will obtain
:
where
is the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
and
is the
vacuum permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
.
Inverse Kepler problem
Consider the inverse Kepler problem. What kind of force law produces a noncircular
elliptical orbit
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 ...
(or more generally a noncircular
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
) around a
focus of the ellipse?
Differentiating twice the above polar equation for an ellipse gives
:
The force law is therefore
:
which is the anticipated inverse square law. Matching the orbital
to physical values like
or
reproduces
Newton's law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
or
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
, respectively.
The effective force for Schwarzschild coordinates is
:
where the second term is an inverse-quartic force corresponding to quadrupole effects such as the angular shift of
periapsis
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary (astronomy), primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two ...
(It can be also obtained via retarded potentials
).
In the
parameterized post-Newtonian formalism
In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order dev ...
we will obtain
:
where
for the
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and
in the classical case.
Cotes spirals
An inverse cube force law has the form
:
The shapes of the orbits of an inverse cube law are known as
Cotes spiral Introduction
In physics and in the mathematics of plane curves, a Cotes's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of spirals classified by Roger Cotes.
Cotes introduces his analysis of these curves as follows: â ...
s. The Binet equation shows that the orbits must be solutions to the equation
:
The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem. When
, the solution is the
epispiral
The epispiral is a plane curve with polar equation
:\ r=a \sec.
There are ''n'' sections if ''n'' is odd and 2''n'' if ''n'' is even.
It is the polar or circle inversion of the rose curve.
In astronomy the epispiral is related to the equations ...
, including the pathological case of a straight line when
. When
, the solution is the
hyperbolic spiral
A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation
:r=\frac
of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called Reciprocal spiral, too..
Pierre ...
. When
the solution is
Poinsot's spiral.
Off-axis circular motion
Although the Binet equation fails to give a unique force law for circular motion about the center of force, the equation can provide a force law when the circle's center and the center of force do not coincide. Consider for example a circular orbit that passes directly through the center of force. A (reciprocal) polar equation for such a circular orbit of diameter
is
:
Differentiating
twice and making use of the
Pythagorean identity
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations be ...
gives
:
The force law is thus
:
Note that solving the general inverse problem, i.e. constructing the orbits of an attractive
force law, is a considerably more difficult problem because it is equivalent to solving
:
which is a second order nonlinear differential equation.
See also
*
*
Classical central-force problem
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, ...
*
General relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
*
Two-body problem in general relativity
The two-body problem in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of lig ...
*
Bertrand's theorem
In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits.
The f ...
References
{{DEFAULTSORT:Binet Equation
Classical mechanics