Binet equation

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

$$F(u^\{-1\})\; =\; -m\; h^2\; u^2\; \backslash left(\backslash frac\{\backslash mathrm\{d\}^\{2\}u\}\{\backslash mathrm\{d\}\backslash theta\; ^2\}+u\backslash right).$$

Newton's second law for a purely central force is

$$F(r)\; =\; m\; \backslash left(\backslash ddot\{r\}-r\backslash dot\{\backslash theta\; \}^2\backslash right).$$

The conservation of angular momentum requires that

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

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

$$\backslash begin\{align\}$$

&\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}

Combining all of the above, we arrive at

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

The general solution is {{Cite book |last=Goldstein|first=Herbert | url=https://www.worldcat.org/oclc/5675073 |title=Classical mechanics |date=1980 |publisher=Addison-Wesley Pub. Co |isbn=0-201-02918-9 |location=Reading, Mass.|oclc=5675073}}

$$\backslash theta\; =\; \backslash int\_\{r\_0\}^r\; \backslash frac\{\backslash mathrm\; dr\}\{r^2\backslash sqrt\{\backslash frac\{2m\}\{L^2\}\; (E-V)\; -\; \backslash frac\{1\}\{r^2\}\}\}\; +\; \backslash theta\_0$$ where $(r\_0,\; \backslash theta\_0)$ is the initial coordinate of the particle.

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

$$-k\; u^2\; =\; -m\; h^2\; u^2\; \backslash left(\backslash frac\{\backslash mathrm\{d\}^\{2\}u\}\{\backslash mathrm\{d\}\backslash theta\; ^\{2\}\}+u\backslash right)$$

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

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

$$l\; u\; =\; 1\; +\; \backslash varepsilon\; \backslash cos\backslash theta.$$

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

The relativistic equation derived for Schwarzschild coordinates is{{Cite web |url=http://www.wbabin.net/science/kren3.pdf |title=Archived copy |access-date=2010-11-15 |archive-url=https://web.archive.org/web/20100619014831/http://wbabin.net/science/kren3.pdf |archive-date=2010-06-19 |url-status=dead }}

$$\backslash frac\{\backslash mathrm\{d\}^\{2\}u\}\{\backslash mathrm\{d\}\backslash theta\; ^\{2\}\}+u=\backslash frac\{r\_s\; c^2\}\{2\; h^\{2\}\}+\backslash frac\{3\; r\_s\}\{2\}u^\{2\}$$

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

$$\backslash frac\{\backslash mathrm\{d\}^\{2\}u\}\{\backslash mathrm\{d\}\backslash theta\; ^\{2\}\}+u=\backslash frac\{r\_s\; c^2\}\{2\; h^2\}+\backslash frac\{3\; r\_s\}\{2\}\; u^2-\backslash frac\{G\; Q^\{2\}\}\{4\; \backslash pi\; \backslash varepsilon\_0\; c^\{4\}\}\backslash left(\backslash frac\{c^2\}\{h^2\}\; u\; +2u^3\backslash right)$$

where $Q$ is the electric charge and $\backslash varepsilon\_0$ is the vacuum permittivity.

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

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

The force law is therefore

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

which is the anticipated inverse square law. Matching the orbital $h^2/l\; =\; \backslash mu$ 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 ishttp://chaos.swarthmore.edu/courses/PDG07/AJP/AJP000352.pdf - The first-order orbital equation

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

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{{Cite arXiv |eprint = astro-ph/0306611|last1 = Behera|first1 = Harihar | title = A flat space-time relativistic explanation for the perihelion advance of Mercury|last2 = Naik|first2 = P. C|year = 2003}}).

In the parameterized post-Newtonian formalism we will obtain

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

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

An inverse cube force law has the form

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

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

$$\backslash frac\{\backslash mathrm\{d\}^2\; u\}\{\backslash mathrm\{d\}\backslash theta^2\}+u=\backslash frac\{k\; u\}\{m\; h^2\}\; =\; C\; u.$$

The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem. When , 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.

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

$$D\; \backslash ,\; u(\backslash theta)=\; \backslash sec\; \backslash theta.$$

Differentiating $u$ twice and making use of the Pythagorean identity gives

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

The force law is thus

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

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

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

which is a second order nonlinear differential equation.

{{Portal|Astronomy|Physics}}

• {{slink|Bohr–Sommerfeld quantization#Relativistic orbit}}
• Classical central-force problem
• General relativity
• Two-body problem in general relativity
• Bertrand's theorem

{{Reflist}}

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Category:Classical mechanics

Category:Eponymous laws of physics