\(Mg_T-(M+m)g>0\)
We assume a linear temperature profile,
\(T=T_{max}(1-\cfrac{x}{d})\)
and a log squared space density profile,
\({ d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(\cfrac { 1 }{ T_{ max }^{ 2 } } ) } ln(\cfrac { 1 }{ T^{ 2 }(x) } )+{ d }_{ n }\)
\({ d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(\cfrac { 1 }{ T_{ max }^{ 2 } } ) } ln(\cfrac { 1 }{ T^2_{max}(1-\cfrac{x}{d})^2 } )+{ d }_{ n }\)
\({ d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(T_{ max }^{ 2 }) } ln(T^{ 2 }_{ max }(1-\cfrac { x }{ d } )^{ 2 })+{ d }_{ n }\)
\( { d }_{ s }(T)=\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(T_{ max }^{ 2 }) } (ln(T^{ 2 }_{ max })+ln((1-\cfrac { x }{ d } )^{ 2 }))+{ d }_{ n }\} \)
\( \cfrac { d(d_{ s }(T)) }{ dx } =\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(T_{ max }^{ 2 }) } (-\cfrac { 1 }{ (1-\cfrac { x }{ d } )^{ 2 } } \cfrac { 2 }{ d } (1-\cfrac { x }{ d } ))\)
\( \cfrac { d(d_{ s }(T)) }{ dx } =\cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(T_{ max }^{ 2 }) } (-\cfrac { 2 }{ (d-x ) } )\)
And we have,
\(g_{ T }=\cfrac { { G }_{ T } }{ { x }_{ o }({ d }_{ n }-{ d }_{ T_{ max } }) } \cfrac { ({ d }_{ T_{ max } }-{ d }_{ n }) }{ ln(T_{ max }^{ 2 }) } (-\cfrac { 2 }{ (d-x) } )\)
\( g_{ T }=\cfrac { 2{ G }_{ T } }{ { x }_{ o }ln(T_{ max }^{ 2 }) }\cfrac { 1}{ (d-x) }\)
\(M\cfrac { 2{ G }_{ T } }{ { x }_{ o }ln(T_{ max }^{ 2 }) } \cfrac { 1}{ d }-(M+m)g>0\)
From this equation we see that small \(x_o\) (radius of body) and small \(d\) provide for greater lift. We shall now optimize \(T_{max}\) for greater lift.
It seem odd that \(g_T\) should be inversely proportional to \(T_{max}\). In free space where \(T_{max}\) is allowed to fall over a distance \(x=L\) to reach \(T=0\) increasing \(T_{max}\) increases \(L\) but \(d_s\) cannot fall below zero. \(\Delta d_s= d_n\) remains fixed, as such,
\(\cfrac{d(d_s(x))}{dx}=\cfrac{d_n}{L}\) when we use a linear approximation, decreases with increasing \(T_{max}\). And so, \(g_T\) decreases.
We can fix \(L\) by placing a cold body at distance \(d\) but still \(\Delta d_s= d_n\) remains the same.
So in practice \(T_{max}\) should be increased until \(g_T\) and so lift is at a maximum and then no further. At this point \(\Delta d_s=d_n\) and cannot be increased further.
