If we start from,
\(T(x)=\cfrac{A}{\sqrt{x}}\)
A body experiences gravitational effect wholly due the mass behind it, the net effect of all mass above the sphere containing the hot mass is zero.
If we introduce the concept of thermal mass, where
\(M_{thermal}=T^4(x)=\left\{\cfrac{A}{\sqrt{x}}\right\}^4\), numerically
We have the total \(M_{thermal}\) behind the body at point \(x\) from the center of \(T\).
\(M_T= \int _{ 0 }^{ x }{ 4\pi x^{ 2 }T^4(x) } dx=C\int _{ 0 }^{ x }{ 4\pi x^{ 2 }\cfrac { 1 }{ x^2 } } dx=C.4\pi x\)
where \(C=A^4\) is a constant.
Then we consider a hot body of total equivalent mass \(C.4\pi x\), what is its thermal gravity outwards? If we consider spheres of area
\(4\pi x^2\)
and the total thermal gravitational flux through them from a unit mass of \(T\), assuming space is empty and nothing else affects the flux between \(x_o\) and \(x\),
\(g_{To}.4\pi x^2_o=g_T.4\pi x^2\)
and that at \(x=x_o\), \(G_{T}=g_{To}.4\pi x^2_o\)
\(g_T=\cfrac{G_{T}}{4\pi x^2}\) per unit mass
\(G_{T}\) increases with equivalent mass by \(C.4\pi x\)
\(g_T=\cfrac{G_{T}}{4\pi x^2}C.4\pi x\)
Let \(G_{To}=G_TC\)
\(g_T=\cfrac{G_{To}}{ x}\)
This gravity is in the direction of \(x\), outwards. This is the result from the post "Thermal Gravity And Lost Innocence" previously, that is to say the equivalent Thermal Mass concept
\(M_{thermal}=T^4(x)\)
assumes that \(g_T=\cfrac{G_{To}}{ x}\). This Thermal Mass generates a positive gravity outwards.
