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I'm reading a book called "Gravity from the ground up" by Bernard Schutz. I don't understand this section from Investigation 13.3 on page 159, which discusses the formation of the solar system:

We saw in Investigation 5.1 on page 43 that the tidal force exerted by a body of mass $M$ on another body at a distance $r$ from it is proportional to $M/r^3$. This in turn is proportional to the mean density of the body, if its mass were spread over the entire region out to $r$.

From this it is easy to see what would happen to a cloud of gas condensing near the Sun. Consider the tidal force of the Sun on the outer part of the condensation, relative to the gravitational force of the condensation itself. This self-force is nothing more than the tidal force of the body on itself, so the two forces are in proportion to the two relevant densities. The tidal force of the Sun is proportional to the mean density of the Sun spread out over the whole region inside the orbit of the condensation. The force of the condensation on itself is proportional to its own density.

Now, in the original collapsing gas cloud from which the Sun and planets formed, the density must have been higher near the center than near the edge. That means that the mean density of the gas inside the orbit of the condensation must have been larger than the density at the condensation. The gas inside went on to form the Sun, but its tidal effect on the condensation didn’t change. Since it had the larger density, its tidal effect dominated. The condensation would therefore have had a very hard time forming.

Specifically, I understand that tidal force is the difference between the gravitational force on one point and the gravitational force on another point. But isn't it proportional to $M/r^3$ only if the two points are near each other? And isn't he talking about two points on opposite sides of the gas cloud, one nearest to the Sun and the other farthest from the Sun? The distance between these two points is not small compared to the radius of the cloud; it's twice its radius. Doesn't the cloud pull each of these points towards itself with a gravitational force proportional to $M/r^2$, not $M/r^3$? And since these forces are in opposite directions, isn't their difference also proportional to $M/r^2$, not $M/r^3$?

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The language of the passage is somewhat confusing, but I think when it talks about "a cloud of gas condensing near the Sun" and subsequently "the condensation" it means a cloud of gas that is small compared to the size of the Sun. And when it talks about "the orbit of the condensation" it is envisaging this small condensing cloud in orbit around the Sun - it does not envisage that the condensing cloud fills its orbit.

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  • $\begingroup$ Yes, I agree. So it's ok to treat the effect of the Sun on the gas cloud as a tidal force proportional to $M/r^3$, where $M$ is the mass of the Sun and $r$ is the distance from the Sun to the cloud, because the size of the cloud is small compared to $r$. But why is it ok to treat the effect of the central portion of the cloud on its outer portion as a tidal force proportional to $M/r^3$, where $M$ is the mass of the cloud and $r$ is the radius of the cloud? I think that's what he's doing, anyway. $\endgroup$
    – user3327311
    Commented Apr 22 at 9:27
  • $\begingroup$ @user3327311 I think $r$ is always the distance from the Sun to the cloud i.e. the radius of the cloud's orbit, not the radius of the cloud itself. I don't think the tidal effect of the cloud on itself is discussed - probably assumed to be not significant. The overall point is that the tidal effect of the sun on the cloud is always greater than the cloud's self-gravitation (as long as the density of the cloud is less than the density of the Sun). $\endgroup$
    – gandalf61
    Commented Apr 22 at 9:34

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