From the post "shhh... A Big Secret...No Wave Overhead, Just In Your Face",
\( \cfrac { dT(t) }{ dt } T(x)=\cfrac { x }{ g_{ T } } \left\{ 2v\cfrac { dv }{ dt } \cfrac { \partial ^{ 2 }T }{ \partial x^{ 2 } } +v^{ 2 }\cfrac { \partial ^{ 2 } }{ \partial x^{ 2 } } (\cfrac { \partial T }{ \partial t } ) \right\} \)
If
\(\cfrac{dv}{dt}=g_T-g\)
where \(g\) is the normal gravity of a massive hot spot, instead of,
\(2xvT(t)\cfrac { d^{ 2 }T(x) }{ dx^{ 2 } } =0\)
we have,
\(2x(\cfrac{g_T-g}{g_T})vT(t)\cfrac { d^{ 2 }T(x) }{ dx^{ 2 } }=0\)
Which means, if
\(g_T-g=0\)
just at the boundary where \(T\) stop accelerating forward, but obviously has some velocity radially away from the hot spot, we can have radial wave behavior (a wave along the radial line) round this boundary.
Beyond this point, \(g\) is higher because \(g_T\) is higher before this zero-cross-over point, otherwise \(T\) would not be propagating outwards, \(T\) emerge decelerating, and would eventually stop under the massive hot spot's gravity pull.
So, there is a distance beyond where the Sun's heat don't reach, but still visible. Bright spot of light but not much heat!
What is even more interesting is that around the boundary where
\(g_T-g=0\)
the conditions are just right for Simple Harmonic Motion along a radial line in and out of this spherical boundary. \(T\) travels beyond this boundary but now decelerating. At the farthest point beyond the boundary \(T\) has zero velocity, but still decelerating under the net effect of \(g\) and \(g_T\) (\(g\) and \(g_T\) both being monotonously decreasing quantities will not cross again Note: This is not strictly true.), races back into the boundary. Inside the boundary it experiences again deceleration because it is now travelling inwards. \(g_T-g\) is positive outwards inside the boundary. \(T\) eventually stops and accelerates outwards, towards the boundary again. SHM.
THIS BOUNDARY IS THE RADIUS OF THE SUN!
The actual mass radius of the Sun is smaller, below this boundary.
Some \(T\) will escape. \(T\) with velocities greater than the Sun's escape velocity will escape from the Sun. This means \(T\) must have a minimum energy to reach us from the Sun. If none of \(T\) have enough escape velocities, the Sun will just be a bright spot in the sky with no heat! The Sun don't have to turn dark when that happens.
This escape velocity is less than the escape velocity of the Sun considering Sun's mass alone. This escape velocity is proportional to the area between the curves, \(g\) and \(g_T\) starting at the boundary and beyond. Its value is expected to be much smaller than just the area under the \(g\) curve.
What if most of \(T\) can escape? There would not be much SHM in and out of the boundary \(g_T-g=0\).
