Kepler Problem

zorop · #1 $Old$ July 26th, 2009, 03:44 PM

The simplified version of the classical one-body "Kepler Problem" [force law given by $f(r)=-\frac{k}{r^2},k>0$ ]. But i want to solve a simplified version of the classical one-body "Coulomb Problem" [force law given by $f(r)=-\frac{k}{r^2},k<0$ ].

(a)Let $\alpha=\frac{L^2}{m|k|}$ show that the polar equation for the orbit is given by $\frac{\alpha}{r}=-1+\varepsilon\cos\theta$ Observe that since both $\alpha$ and $r$ are positive, no real orbits exist unless $\varepsilon>C$ , where C is a positive constant. What must C be? What then do you conclude about the orbits?

(b)What is $\varepsilon$ (in terms of the physical parameters m, k, E and L)?

(c)Show that the orbit equation, in Cartesian coordinates, is given by $Ax^2+By^2+Cx+Dy+F=0$ and determine explicitly the constants A, B, C, D, and F in terms of $\alpha$ and $\varepsilon$ .

(d)What types of orbits are possible for the Coulomb problem?

(e)Find the distance of closest approach $r_{min}$ to the center of force (in terms of $\alpha$ and $\varepsilon$ ).

(f)Show that the major axis of the orbit is independent of L (unlike the minor axis).

Thanks very much~~~

luobo · #2 $Old$ August 10th, 2009, 08:30 PM

Quote:

Originally Posted by zorop $View Post$

The simplified version of the classical one-body "Kepler Problem" [force law given by $f(r)=-\frac{k}{r^2},k>0$ ]. But i want to solve a simplified version of the classical one-body "Coulomb Problem" [force law given by $f(r)=-\frac{k}{r^2},k<0$ ].

(a)Let $\alpha=\frac{L^2}{m|k|}$ show that the polar equation for the orbit is given by $\frac{\alpha}{r}=-1+\varepsilon\cos\theta$ Observe that since both $\alpha$ and $r$ are positive, no real orbits exist unless $\varepsilon>C$ , where C is a positive constant. What must C be? What then do you conclude about the orbits?

(b)What is $\varepsilon$ (in terms of the physical parameters m, k, E and L)?

(c)Show that the orbit equation, in Cartesian coordinates, is given by $Ax^2+By^2+Cx+Dy+F=0$ and determine explicitly the constants A, B, C, D, and F in terms of $\alpha$ and $\varepsilon$ .

(d)What types of orbits are possible for the Coulomb problem?

(e)Find the distance of closest approach $r_{min}$ to the center of force (in terms of $\alpha$ and $\varepsilon$ ).

(f)Show that the major axis of the orbit is independent of L (unlike the minor axis).

Thanks very much~~~

(a)
Equation of the Orbit:
$m\ddot{r}=-\frac{k}{r^2}+\frac{L^2}{mr^3} (k<0)$ (1)
(The following can be found in many text books.)
Let $u=\frac{1}{r}$ and rewrite the equation (1) in terms of $\theta$ rather than time $t$ , then
$\frac{d^2u}{d\theta^2}+u=\frac{mk}{L^2}=-\frac{1}{\alpha}$ (2)

Solution of (2) is
$\frac{1}{r}=u=-\frac{1}{\alpha}+A\cos(\theta-\theta_0)$ (3)

Therefore,
$\frac{\alpha}{r}=-1+\alpha A\cos(\theta-\theta_0)$ (4)

Let $\theta-\theta_0\rightarrow\theta, e=\alpha A$ ,
$\frac{\alpha}{r}=-1+e\cos\theta$ (5)

$C$ must be larger than or equal to 1. The orbit is unbounded.

(b)(c)(d)(e)(f) are trivial.