Thursday, January 23, 2014

This is another post about "Gravitational Pull"

As I closed last time, "gravitational pull" tends not to produce collisions — at least, not in idealized circumstances, which I described with the phrase "essentially-two". In the old, "classical" physics setting, this is due to the particular (empirically-determined, but excellently predictive!) gravitational potential, which has a large group of symmetries. The model of a two-object interaction then looks like
\[ \frac{1}{2} \left( \frac{d}{dt} r \right)^2 + \frac{1}{2} r^2 \left(\frac{d}{dt} \Theta\right)^2 - \frac{g_0}{r} = H \qquad\mbox{is constant}\]
where $r$ is the (variable) distance between the two objects and $\Theta$ is a cleverly-composed something that describes the (variable) direction of the two objects' separation, and $g_0$ is a constant describing the gravitational attraction between the two things. It happens that another expression
\[ r \Theta \times r \frac{d}{dt} \Theta = L \] must also be a constant — this one having directional information.

The never-meeting of the two things is sumarized in the necessity that squares of real things be positive:
\[ H + \frac{g_0}{r} - \frac{1}{2r^2} L^2 = \frac{1}{2} \left( \frac{d}{dt} r \right) ^2 \geq 0 \]

Now, of course we know that things largely under the influence of gravity do occasionally hit one another, and so the Moon is a rather interesting thing to behold at night, and St. Laurence has his “tears” every year — these are instances of things having size, being more than geometric points — and so of not being “essentially-two”, in my strange turn of phrase. Another class of things exerting mutual gravitational pull that aren't “essentially-two” in this sense is that of binary stellar systems. A notable sub-class are the pairings of a red giant and a white dwarf, in relative proximity. Sometimes these sorts of collisions take the form of the dwarf partner gradually accumulating the loose hydrogen from off their red giant partner, untill it gets thick and hot enough to start fusing hydrogen into helium on its own account, which makes a kind of nova. I wonder if (and how) this picture might have been in Benedict's analogical thought when he spoke or wrote of "gravitational pull"?


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