28 KiB
| title |
|---|
| 15 Kepler's problem |
An important class of central fields is formed by those in which the potential energy is inversely proportional to r, and the force accordingly inversely proportional to r^2. They include the fields of Newtonian gravitational attraction and of Coulomb electrostatic interaction; the latter may be either attractive or repulsive.
Let us first consider an attractive field, where
15.1
with a a positive constant. The "effective" potential energy
15.2
is of the form shown in fig10. As r\to 0, U_\text{eff} tends to +\infty, and as r\to \infty it tends to zero from negative values ; for r = M^2/m\alpha it has a minimum value
15.3
10
It is seen at once from fig10 that the motion is finite for E <0 and infinite for E > 0.
The shape of the path is obtained from the general formula 14.7. Substituting there U=-\alpha/r and effecting the elementary integration, we have
\phi=\cos^{-1}\frac{(M/r)-(m\alpha/M)}{\sqrt{2mE+\frac{m^2\alpha^2}{M^2}}}
+\constant
Taking the origin of such that the constant is zero, and putting
15.4
we can write the equation of the path as
15.5
This is the equation of a conic section with one focus at the origin; 2p is called the latus rectum of the orbit and e the eccentricity. Our choice of the origin of is seen from 15.5 to be such that the point where \phi = 0 is the point nearest to the origin (called the perihelion).
In the equivalent problem of two particles interacting according to the law 15.1, the orbit of each particl$e$ is a conic section, with one focus at the centre of mass of the two particles.
It is seen from 15.4 that, if E\lt 0, then the eccentricity e \lt 1, i.e. the orbit is an ellipse fig11 and the motion is finite, in accordance with what has been said earlier in this section. According to the formulae of analytical geometry, the major and minor semi-axes of the ellipse are
15.6
11
The least possible value of the energy is 15.3, and then e = 0, i.e. the ellipse becomes a circle. It may be noted that the major axis of the ellipse depends only on the energy of the particle, and not on its angular momentum. The least and greatest distances from the centre of the field (the focus of the ellipse) are
15.7
These expressions, with a and e given by 15.6 and 15.4, can, of course, also be obtained directly as the roots of the equation U_\text{eff}(r) = E.
The period T of revolution in an elliptical orbit is conveniently found by using the law of conservation of angular momentum in the form of the area integral 14.3. Integrating this equation with respect to time from zero to T, we have 2mf = TM, where f is the area of the orbit. For an ellipse f = \pi ab, and by using the formulae 15.6 we find
15.8
The proportionality between the square of the period and the cube of the linear dimension of the orbit has already been demonstrated in 10. It may also be noted that the period depends only on the energy of the particle.
For E \gt 0 the motion is infinite. If E > 0, the eccentricity e > 1, i.e. the the path is a hyperbola with the origin as internal focus fig12. The distance of the perihelion from the focus is
15.9
where a = p/(e^2-1) = \alpha/2E is the "semi-axis" of the hyperbola.
12
If E = 0, the eccentricity e = 1, and the particle moves in a parabola with perihelion distance r_\min = \mfrac{1}{2}p. This case occurs if the particle starts from rest at infinity.
The co-ordinates of the particle as functions of time in the orbit may be found by means of the general formula 14.6. They may be represented in a convenient parametric form as follows.
Let us first consider elliptical orbits. With a and e given by 15.6 and 15.4 we can write the integral 14.6 for the time as
\begin{align} t &=\sqrt{\frac{m}{2|E|}}\int\frac{r\dd{r}}{\sqrt{-r^2+(\alpha/|E|)r-(M^2/2m|E|)}} \ &=\sqrt{\frac{ma}{\alpha}}\int\frac{r\dd{r}}{\sqrt{a^2e^2-(r-a)^2}} \end{align}
The obvious substitution r-a =-ae\cos\xi converts the integral to
t=\sqrt{\frac{ma^3}{\alpha}}\int(1-e\cos\xi)\dd{\xi}=\sqrt{\frac{ma^3}{\alpha}}(\xi-e\sin\xi)+\constant
If time is measured in such a way that the constant is zero, we have the
following parametric dependence of r on t:
15.10
the particle being at perihelion at t = 0. The Cartesian co-ordinates x = r\cos\phi, y = r \sin\phi (the x and y axes being respectively parallel to the major and minor axes of the ellipse) can likewise be expressed in terms of the parameter \xi. From 15.5 and 15.10 we have
ex=p-r=a(1-e^2)-a(1-e\cos\xi)=ae(\cos\xi-e)
y is equal to \sqrt{r^2-x^2}. Thus
15.11
A complete passage round the ellipse corresponds to an increase of \xi from 0 to 2\pi.
Entirely similar calculations for the hyperbolic orbits give
15.12
where the parameter \xi varies from -\infty to +\infty.
Let us now consider motion in a repulsive field, where
15.13
Here the effective potential energy is
U_\text{eff}=\frac{\alpha}{r}+\frac{M^2}{2mr^2}
and decreases monotonically from +\infty to zero as r varies from zero to infinity. The energy of the particle must be positive, and the motion is always infinite. The calculations are exactly similar to those for the attractive field. The path is a hyperbola (or, if E = 0, a parabola):
15.14
where p and e are again given by 15.4. The path passes the centre of the field in the manner shown in fig13 The perihelion distance is
15.15
The time dependence is given by the parametric equations
15.16
To conclude this section, we shall show that there is an integral of the motion which exists only in fields U = \alpha/r (with either sign of \alpha). It is easy to verify by direct calculation that the quantity
15.17
is constant. For its total time derivative is \dot{\v{v}}\times\v{M}+\alpha\v{v}/r-\alpha\v{r}(\v{v}\cdot\v{r})/r^3 or,
m\v{r}(\v{v}\cdot\dot{\v{v}})-m\v{v}(\v{r}\cdot\dot{\v{v}})+\alpha\v{v}/r-\alpha\v{r}(\v{r}\cdot\v{r})/r^3
Putting m\dot{\v{v}} = \alpha\v{r}/r^3 from the equation of motion, we find that this expression vanishes.
13
The direction of the conserved vector 15.17 is along the major axis from the focus to the perihelion, and its magnitude is \alpha e. This is most simply seen by considering its value at perihelion.
It should be emphasised that the integral 15.17 of the motion, like M and E, is a one-valued function of the state (position and velocity) of the particle. We shall see in 50 that the existence of such a further one-valued integral is due to the degeneracy of the motion.