Burning Sun

The post "Temperature", which served well and led to many possibilities, however is too assuming.  Specifically the expression that led to the sign before the gravity term to flip is,

$kg = \rho X_A x$

which suggests that, if we apply the analogy now,

$T(x) = \rho X_A x$

Strictly speaking this is not a valid use of analogy.  In normal application, all terms are reduced first before a replacement is made.   The flip in sign for the gravity term result as

$T(x) = T_{max}-\rho X_A x$

in the target analogous case.   Temperature decreases with distance  $x$  from the hot spot.  This is why $-g_T$  was used.  The point is,  $T$  is not linear in  $x$ throughout the distance  $L$.  However, for the first part of the  $g_T$  curve just before the intersection with  $g$,  the curve can be approximated as a straight line.  All expressions derived based on a linear dependence on  $x$  are still valid up to boundary  $g_T-g=0$.

Once the temperature profile is established,  between a hot spot at $T=T_{max}$  and some point  $x=L$ where  $T=0$,  it does not change with time.

There is however a $\cfrac{dT}{dt}$  term, that indicates the flow of  $T$  down the temperature profile.  Once the temperature profile is established, however,

 $\cfrac{dT}{dt}=constant$

If,

 $\cfrac{dT}{dt}\rightarrow\infty$

then the temperature profile will keep growing,  $L$ reaching further out into space.

But does  $T$  behave like a wave while its flows?  Post "Heat Wave" and "shhh... A Big Secret...No Wave Overhead, Just In Your Face"  try to find a wave travelling radially down to establish the temperature profile.  The former post approximated for  $T(x)$  and showed that  $g_T(x)$, the field equation for thermal gravity is needed.  A case of Chicken needing an Egg...  The latter post suggests that there is no radial wave assuming that  $g_T$ is high and that the hot spot gravity is ignored.  The post "Where the Sun Don't Shine And The Chill Cold Space" considered the hot spot gravity and finds that there may be a wave along the radial line near when $g_T-g=0$.  $T$ seem to perform SHM in and out of this boundary.  The gravity reversal at this zero points caused gravity bands to form around the hot spot.  The radial wave that exist over this short region will have the same form as the concentric wave from the post "Heat Wave Again", except  $x$  is now along the radius.  It was proposed in that post that once a temperature profile has been established, concentric waves travel around the hot spot and multiple concentric waves extends all the way till  $T=0$  when  $x=L$.

$\cfrac{dT}{dt}$  exist up till  $x$  reaches  $L$.  Once  $x$ reaches  $L$,  $T$ stops.  Like a hot body that melts because of its temperature, it collapses but eventually stops deforming.   Even if there is a standing wave between  $T=T_{max}$  and  $T=0$, it does not deliver a net flow of   $T$.

So the Sun is just a hot spot with an established  $T$  profile,  finite  $\cfrac{dT}{dt}$  is needed from it.  The Sun will last a long time,  it is not required to burn fuel at an infinite rate.  All that is required of the Sun is just to be hot.  Very HOT.