\(T(x) = \cfrac{A}{\sqrt{x}}=A(x)^{-\cfrac{1}{2}}\)
\(A\) is a constant of proportionality. Differentiating with respect to \(x\),
\({ T }^{ ' }(x)\quad =-\cfrac { A }{ 2 } { (x) }^{ -\cfrac { 3 }{ 2 } }\)
\( { T }^{ '' }(x)\quad =\cfrac { 3A }{ 4 } { (x) }^{ -\cfrac { 5 }{ 2 } }\)
and from the post "What Thermal Gravity? Too Hot To Handle..."
\( g_{ T } =v^2\cfrac{x}{T(x)}\cfrac { d^{ 2 }T(x) }{ d x^{ 2 } } \)
\( g_{ T } =v^2\cfrac{x(x)^\cfrac{1}{2}}{A}\cfrac { 3A }{ 4 } { (x) }^{ -\cfrac { 5 }{ 2 } } \)
\(g_{ T }=\cfrac { 3 }{ 4x } v^{ 2 }=\cfrac{a}{x}=S_TT^2\)
where \(v\) is the speed of \(T\) as it travels across the temperature profile, \(a\) a constant, and \(S_T\) a constant of proportionality with the unit ms-2J-2. The last equation relates \(g_T\) directly to \(T\) this will bring us to a relationship for space density and temperature in another post.
Unfortunately, a plot of a family of curves of \(\cfrac{a}{x}\) together with \(\cfrac{1}{x^2}\),
shows that \(g_T={a}/{x}\) does not intersect \(g={1}/{x^2}\) again after its goes above \({1}/{x^2}\) ie after \(g_T>g\) along \(x\). \(g_T\) has to be greater than \(g\) at the start \(x=x_o\) for \(T\) to flow. This is the case for nebula where the boundary, \(g_T-g=0\) does not exist, but not a general case of planetary spheres and rings where \(g_T>g\) for \(T\) to flow outwards and then \(g_T-g=0\) at some distance \(x_{ring}\) beyond which \(g_T\) oscillate up and down \(g\). It is not appropriate to repesent gravity of the hot spot here using \(1/x^2\).
However, if instead we use \(e^{-x}\) to represent gravity,
Both Nebula and planetary spheres and rings scenarios are encountered as \({1}/{x}\) is scaled appropriately. The curve marked as \({a}/{x}\) is higher than the gravity curve \(g\) and does not intersect \(g\). This is the Nebula scenario where \(T\) spreads over vast distances.
All other curves below \({a}/{x}\) cross \(g\) twice. The second intersection further out along \(x\) marks a gap where the net gravity due to \(g\) and \(g_T\) points away from it, before and after that point.
It should be noted that, beyond the first intersection, \(T\) redistributes and \(g_T\) is no longer of the form \({a}/{x}\) but is expected to have a sinusoidal component and intersect \(g\) again multiple times.
Below is a plot of (1/x)*cos(2*x)^2/(cos(2)^2) to show how \(g_T\) might look like,
If \(T(x)\) can be determined from observations, then logically \(g_T\) should come after \(T(x)\) has been formulated. We shall do that in the next post.
\(g_T\) was derived from an expression for \(T\) temperature, based on an moving mass analogy for \(T\), together with the assumption that \(T\) propagates as a wave, at a constant velocity \(v\). If \(T(x)\) can be obtained empirically, then \(g_T\) can be derived directly with just one assumption of how \(T\) affects \(g_T\) through interacting with space.
