Binet equation

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

The shape of an orbit is often conveniently described in terms of relative distance ${\displaystyle r}$ as a function of angle ${\displaystyle \theta }$ . For the Binet equation, the orbital shape is instead more concisely described by the reciprocal ${\displaystyle u=1/r}$ as a function of ${\displaystyle \theta }$. Define the specific angular momentum as ${\displaystyle h=L/m}$ where ${\displaystyle L}$ is the angular momentum and ${\displaystyle m}$ is the mass. The Binet equation, derived in the next section, gives the force in terms of the function ${\displaystyle u(\theta )}$:

${\displaystyle F({u}^{-1})=-mh^{2}u^{2}\left({\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u\right).}$

Derivation

Newton's Second Law for a purely central force is

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

The conservation of angular momentum requires that

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

Derivatives of ${\displaystyle r}$ with respect to time may be rewritten as derivatives of ${\displaystyle u=1/r}$ with respect to angle:

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

Combining all of the above, we arrive at

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

Examples

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

${\displaystyle -ku^{2}=-mh^{2}u^{2}\left({\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u\right)}$

${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {k}{mh^{2}}}\equiv {\text{constant}}>0.}$

If the angle ${\displaystyle \theta }$ is measured from the periapsis, then the general solution for the orbit expressed in (reciprocal) polar coordinates is

${\displaystyle lu=1+\varepsilon \cos \theta .}$

The above polar equation describes conic sections, with ${\displaystyle l}$ the semi-latus rectum (equal to ${\displaystyle h^{2}/\mu =h^{2}m/k}$) and ${\displaystyle \varepsilon }$ the orbital eccentricity.

Relativistic

The relativistic equation derived for Schwarzschild coordinates is[1]

${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {r_{s}c^{2}}{2h^{2}}}+{\frac {3r_{s}}{2}}u^{2}}$

where ${\displaystyle c}$ is the speed of light and ${\displaystyle r_{s}}$ is the Schwarzschild radius. And for Reissner–Nordström metric we will obtain

${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {r_{s}c^{2}}{2h^{2}}}+{\frac {3r_{s}}{2}}u^{2}-{\frac {GQ^{2}}{4\pi \varepsilon _{0}c^{4}}}\left({\frac {c^{2}}{h^{2}}}u+2u^{3}\right)}$

where ${\displaystyle Q}$ is the electric charge and ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity.

Inverse Kepler problem

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

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

The force law is therefore

${\displaystyle F=-mh^{2}u^{2}\left({\frac {-\varepsilon \cos \theta }{l}}+{\frac {1+\varepsilon \cos \theta }{l}}\right)=-{\frac {mh^{2}u^{2}}{l}}=-{\frac {mh^{2}}{lr^{2}}},}$

which is the anticipated inverse square law. Matching the orbital ${\displaystyle h^{2}/l=\mu }$ to physical values like ${\displaystyle GM}$ or ${\displaystyle k_{e}q_{1}q_{2}/m}$ reproduces Newton's law of universal gravitation or Coulomb's law, respectively.

The effective force for Schwarzschild coordinates is[2]

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

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[3]).

In the parameterized post-Newtonian formalism we will obtain

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

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

Cotes spirals

An inverse cube force law has the form

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

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

${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {ku}{mh^{2}}}=Cu.}$

The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem. When ${\displaystyle C<1}$, the solution is the epispiral, including the pathological case of a straight line when ${\displaystyle C=0}$. When ${\displaystyle C=1}$, the solution is the hyperbolic spiral. When ${\displaystyle C>1}$ 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 ${\displaystyle D}$ is

${\displaystyle D\,u(\theta )=\sec \theta .}$

Differentiating ${\displaystyle u}$ twice and making use of the Pythagorean identity gives

${\displaystyle D\,{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}=\sec \theta \tan ^{2}\theta +\sec ^{3}\theta =\sec \theta (\sec ^{2}\theta -1)+\sec ^{3}\theta =2D^{3}u^{3}-D\,u.}$

The force law is thus

${\displaystyle F=-mh^{2}u^{2}\left(2D^{2}u^{3}-u+u\right)=-2mh^{2}D^{2}u^{5}=-{\frac {2mh^{2}D^{2}}{r^{5}}}.}$

Note that solving the general inverse problem, i.e. constructing the orbits of an attractive ${\displaystyle 1/r^{5}}$ force law, is a considerably more difficult problem because it is equivalent to solving

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

which is a second order nonlinear differential equation.

See also

• Bohr–Sommerfeld quantization#Relativistic orbit
• Classical central-force problem
• General relativity
• Two-body problem in general relativity
• Bertrand theorem

References

1. "Archived copy" (PDF). Archived from the original (PDF) on 2010-06-19. Retrieved 2010-11-15.CS1 maint: archived copy as title (link)
2. http://chaos.swarthmore.edu/courses/PDG07/AJP/AJP000352.pdf - The first-order orbital equation
3. Behera, Harihar; Naik, P. C (2003). "A flat space-time relativistic explanation for the perihelion advance of Mercury". arXiv:astro-ph/0306611.