The Binet equation, derived by
Jacques Philippe Marie Binet, provides the form of a
central force given the shape of the
orbital motion
In celestial mechanics, an orbit (also known as orbital revolution) 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 obj ...
in plane
polar coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are
*the point's distance from a reference ...
. 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,
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
. A unique solution is impossible in the case of
circular motion 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 as
where
is the
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
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 physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
# A body re ...
for a purely central force is
The
conservation of angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
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 of calculating the orbit of an
inverse square law 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 body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values.
Apsides perta ...
, then the general solution for the orbit expressed in (reciprocal) polar coordinates is
The above polar equation describes
conic section
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
s, with
the
semi-latus rectum (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 be ...
.
Relativistic
The relativistic equation derived for
Schwarzschild coordinates is
where
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
and
is the
Schwarzschild radius. And for
Reissner–Nordström metric we will obtain
where
is the
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
and
is the
vacuum permittivity.
Inverse Kepler problem
Consider the inverse Kepler problem. What kind of force law produces a noncircular
elliptical orbit
In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an orbital eccentricity, eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some or ...
(or more generally a noncircular
conic section
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
) 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 describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is Proportionality (mathematics)#Direct proportionality, proportional to the product ...
or
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
, 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 body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values.
Apsides perta ...
(It can be also obtained via retarded potentials
).
In the
parameterized post-Newtonian formalism we will obtain
where
for the
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
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 spirals. 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, including the pathological case of a straight line when
. When
, the solution is the
hyperbolic spiral. 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 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
*
General relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
*
Two-body problem in general relativity
*
Bertrand's theorem
References
{{DEFAULTSORT:Binet Equation
Classical mechanics
Eponymous laws of physics