Binet equation

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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 plane polar coordinates. 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. A unique solution is impossible in the case of circular motion about the center of force.

Equation[edit]

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 and m is the mass. The Binet equation[1] is

F(u)=-mh^{2}u^{2}\left(\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u\right).

Derivation[edit]

Newton's Second Law for a purely central force is

F(r)=m(\ddot{r}-r\dot{\theta }^{2}).

The conservation of angular momentum requires that

r^{2}\dot{\theta }=h=\text{constant}.

Derivatives of r with respect to time may be rewritten as derivatives of u with respect to angle.

\begin{align}
 &\frac{\mathrm{d}u}{\mathrm{d}\theta }=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{r}\right)\frac{\mathrm{d}t}{\mathrm{d}\theta }=-\frac{{\dot{r}}}{r^{2}\dot{\theta }}=-\frac{{\dot{r}}}{h} \\ 
 & \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}=-\frac{1}{h}\frac{\mathrm{d}\dot{r}}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\theta }=-\frac{{\ddot{r}}}{h\dot{\theta }}=-\frac{{\ddot{r}}}{h^{2}u^{2}} \\
\end{align}

Combine all of the above and we have

F=m(\ddot{r}-r\dot{\theta }^2)=-m\left(h^{2}u^{2}\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+h^{2}u^{3}\right)=-mh^{2}u^{2}\left(\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u\right)

Examples[edit]

Kepler problem[edit]

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

\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u=\text{constant}>0.

If the angle \theta is measured from the periapsis, 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 sections, with l the semi-latus rectum and \varepsilon the orbital eccentricity.

The relativistic equation derived for Schwarzschild coordinates is[2]

\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u=\frac{r_s c^{2}}{2 h^{2}}+\frac{3 r_s}{2}u^{2}

where c is the speed of light and r_s is the Schwarzschild radius. And for Reissner–Nordström metric we will obtain

\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u=\frac{r_s c^{2}}{2 h^{2}}+\frac{3 r_s}{2}u^{2}-\frac{G Q^{2}}{4 \pi \varepsilon_0 c^{4}}\left(\frac{c^{2}}{h^{2}}u+2u^3\right)

where Q is the electric charge and \varepsilon_0 is the vacuum permittivity.

Inverse Kepler problem[edit]

Consider the inverse Kepler problem. What kind of force law produces a noncircular elliptical orbit (or more generally a noncircular conic section) around a focus of the ellipse?

Differentiating twice the above polar equation for an ellipse gives

l \, \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}} = - \varepsilon \cos \theta.

The force law is therefore

F=-mh^{2}u^{2}\left(\frac{- \varepsilon \cos \theta}{l}+\frac{1 + \varepsilon \cos \theta}{l}\right)=-\frac{m h^2 u^2}{l}=-\frac{m h^2}{l r^2},

which is the anticipated inverse square law. Matching the orbital h^2/l to physical values like GM or k_e q_1 q_2/m reproduces Newton's law of universal gravitation or Coulomb's law, respectively.

The effective force for Schwarzschild coordinates is[3]

F=-GMmu^{2}\left(1+3\left(\frac{hu}{c}\right)^{2}\right)=-\frac{GMm}{r^{2}}\left(1+3\left(\frac{h}{rc}\right)^{2}\right).

where the second term is an inverse-quartic force corresponding to quadrupole effects such as the angular shift of periapsis (It can be also obtained via retarded potentials[4]).

In the parameterized post-Newtonian formalism we will obtain

F=-\frac{GMm}{r^{2}}\left(1+(2+2\gamma-\beta)\left(\frac{h}{rc}\right)^{2}\right).

where \gamma=\beta=1 for the general relativity and \gamma=\beta=0 in the classical case.

Cotes spirals[edit]

An inverse cube force law has the form

F(r)=-\frac{k}{r^3}.

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{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u=\frac{k u}{m h^2} = 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[edit]

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{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}=\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{2mh^2D^2}{r^5}.

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{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u=Cu^3

which is a second order nonlinear differential equation.

See also[edit]

References[edit]

  1. ^ "Fyta12:1 – Motion in a Central Force Field". Retrieved 2010-10-05. 
  2. ^ http://www.wbabin.net/science/kren3.pdf
  3. ^ http://chaos.swarthmore.edu/courses/PDG07/AJP/AJP000352.pdf - The first-order orbital equation
  4. ^ http://arxiv.org/pdf/astro-ph/0306611.pdf - A flat space-time relativistic explanation for the perihelion advance of Mercury

External links[edit]