Gravity inequalities and the mean temperature on a planet

Abstract

Abstract In the centered surround system S ( 2 ) { P , Γ } $S^{(2)} \{P,\varGamma \}$ , we establish the following gravity inequalities: ( 1 + 5 2 π e 2 1 − e 2 ) ( 2 π | Γ | ) 2 ≤ 1 | Γ | ∮ Γ 1 ∥ A − P ∥ 2 ≤ ( 1 + 16 − π 4 π e 2 1 − e 2 ) ( 2 π | Γ | ) 2 , $$\biggl(1+\frac{5}{2\pi}\frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{2}\leq\frac{1}{|\varGamma |} \oint_{\varGamma }\frac{1}{\|A-P\|^{2}} \leq\biggl(1+\frac{16-\pi}{4\pi} \frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi }{|\varGamma |} \biggr)^{2}, $$ where Γ is an ellipse, P and e are one of the foci and the eccentricity of the ellipse, respectively, and A ∈ Γ $A\in \varGamma $ is a satellite of the centered surround system S ( 2 ) { P , Γ } $S^{(2)} \{P,\varGamma \}$ . We also demonstrate the applications of the inequalities in the temperature research, and we obtain an approximate mean temperature formula as follows: T ‾ ≈ ( 1 + 0.9095071300973198 … × e 2 1 − e 2 ) ( 2 π | Γ | ) 2 , ∀ e ∈ ( 0 , 1 ) , $$\overline{T}\approx\biggl(1+0.9095071300973198\ldots\times\frac{\mathrm {e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{2}, \quad\forall\mathrm{e} \in(0,1), $$ where the T̅ is the mean temperature on a planet.

Keywords

• centered surround system
• mean gravity norm
• isoperimetric inequality
• Newton formula
• Euler formula