Thermal Gravity And Lost Innocence

From the web,  The surface temperature of planet as a function of   $x$  distance from the Sun is given by,

$T(x) = \cfrac{A}{\sqrt{x}}=A(x)^{-\cfrac{1}{2}}$

$A$  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.