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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 r as a function of angle \theta. For the Binet equation, the orbital shape is instead more concisely described by the reciprocal u = 1/r as a function of \theta. Define the specific angular momentum as h=L/m where L 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 m is the mass. The Binet equation, derived in the next section, gives the force in terms of the function u(\theta) : F(u^) = -m h^2 u^2 \left(\frac+u\right).


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 F(r) = m \left(\ddot-r\dot^2\right). 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 r^\dot = h = \text. Derivatives of r with respect to time may be rewritten as derivatives of u=1/r with respect to angle: \begin &\frac = \frac\left(\frac\right)\frac=-\frac=-\frac \\ & \frac=-\frac\frac\frac=-\frac = -\frac \end Combining all of the above, we arrive at F = m\left(\ddot-r\dot^2\right) = -m\left(h^2 u^2 \frac +h^u^\right)=-mh^u^\left(\frac+u\right) The general solution is \theta = \int_^r \frac + \theta_0 where (r_0, \theta_0) 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 -k u^2 = -m h^2 u^2 \left(\frac+u\right) \frac+u = \frac \equiv \text>0. If the angle \theta 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 l u = 1 + \varepsilon \cos\theta. 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 l the semi-latus rectum (equal to h^2/\mu = h^2m/k) and \varepsilon 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 \frac+u=\frac+\fracu^ where c 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 r_s is the Schwarzschild radius. And for Reissner–Nordström metric we will obtain \frac+u=\frac+\frac u^2-\frac\left(\frac u +2u^3\right) where Q 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 \varepsilon_0 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 l \, \frac = - \varepsilon \cos \theta. The force law is therefore F = -mh^u^ \left(\frac+\frac\right)=-\frac=-\frac, which is the anticipated inverse square law. Matching the orbital h^2/l = \mu to physical values like GM or k_e q_1 q_2/m 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 F = -GMmu^2 \left(1+3\left(\frac\right)^2\right)= - \frac \left(1+3\left(\frac\right)^2\right). 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 F = -\frac \left(1+(2+2\gamma-\beta)\left(\frac\right)^2\right). where \gamma = \beta = 1 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 \gamma = \beta = 0 in the classical case.


Cotes spirals

An inverse cube force law has the form F(r) = -\frac. 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 \frac+u=\frac = C u. The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem. When C < 1, the solution is the epispiral, including the pathological case of a straight line when C = 0. When C = 1, the solution is the hyperbolic spiral. When C > 1 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 D is D \, u(\theta)= \sec \theta. Differentiating u twice and making use of the Pythagorean identity gives D \, \frac = \sec \theta \tan^2 \theta + \sec^3 \theta = \sec \theta (\sec^2 \theta - 1) + \sec^3 \theta = 2 D^3 u^3-D \, u. The force law is thus F = -mh^2u^2 \left( 2 D^2 u^3- u + u\right) = -2mh^2D^2u^5 = -\frac. Note that solving the general inverse problem, i.e. constructing the orbits of an attractive 1/r^5 force law, is a considerably more difficult problem because it is equivalent to solving \frac+u=Cu^3 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