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Why does Goldstein's derivation of Keplerian orbits from Newton's law of gravity not yield the orbit consisting of symmetric oscillation along a line passing thru the center of the Sun?

The fact that an orbiting body cannot penetrate the sun is irrelevant here. By Newton's theory, the oscillating body should accelerate to infinite velocity as it approaches the center, but then immediately decelerate symmetrically on the other side. The period will obviously be finite. This orbit is a maximally eccentric ellipse, and its foci are at the ends, but the sun is at the center!

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  • $\begingroup$ So the sun would not be at a focus? $\endgroup$ – BowlOfRed Nov 10 '15 at 5:49
  • $\begingroup$ Goldstein talks about stable orbits not passing through the centre of force when he derives keplerian orbits. When he draws the graph of potential vs r, the bottom most point of the valley is the keplerian orbit. But depending upon the energy of the body and initial position, other orbits are possible. One of which is this. $\endgroup$ – sarat.kant Nov 11 '15 at 17:36
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There are two reasons.

Inside the Sun, the force is no longer an inverse-square law. It actually grows linearly with $r$.

The second reason is that Goldstein (as well as any other classical mechanics book) is interested in orbits with a non vanishing angular momentum with respect to the center of the Sun. An oscillation along a line passing through this point has zero angular momentum. We cannot think of this orbit as the limiting case of the ellipse. The latter has eccentricity $$\epsilon=\sqrt{1+\frac{2EL^2}{mK^2}},$$ where $L$ is the angular momentum. By doing $L\rightarrow 0$ one gets $\epsilon\rightarrow 1$ which is a parabola (which does not correspond to an oscillation). This is inconsistent because in obtaining the eccentricity it is already assumed that $L\neq 0$.

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Actually it is similar to the solution of a body going through the center of the earth through a drilled hole .

Neutrinos could go through such an orbit with zero angular momentum as the probability of interacting and disappearing is very small. Planets cannot. If Goldstein is in a chapter for planetary orbits (hint Keplerian) it is obvious why the one dimensional solution is not considered, it would not be an orbit but an impact.

After all, if one took the elliptic solution to the limit of zero small axis one would get your solution after all.

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