1 Introduction
It is well known that there are eight planets in the solar system, i.e., Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. In space science, we are concerned with the radiation energy of the Sun since it will directly affect our daily life. Since the temperature on a planet is dependent on the radiation energy and the radiation energy is related to the gravity of the Sun, we are especially interested in the gravity norm of the Sun.
In space science, we need to know the mean temperature on a planet. Unfortunately, it is very difficult to measure the mean temperature on a planet. Therefore, it is of theoretical significance to study the mean temperature on a planet by means of mathematics.
At present, the research of temperature on the Earth is a topic of focus in the world. In our daily life, we are concerned with temperature change, which will directly affect our daily life. Since the rain and the air humidity are related to the temperature and our daily life is dependent on the rain and the air humidity, it is of value in applications to study the mean temperature on the Earth.
In this paper, our motivation is to study the mean temperature on a planet by means of the theory of mean [
1]. To this end, we first introduce the basic concepts in the surround system [
2‐
5], and we illustrate the background and the significance of these concepts in space science. Next, we establish several identities and inequalities involving the centered surround system
\(S^{(2)} \{ P,\varGamma \}\), in particular, the
mean gravity norm formula, as well as we illustrate the coefficients in these inequalities are the best constants. Next, we prove
gravity inequalities in the centered surround system
\(S^{(2)} \{P,\varGamma \}\), which are also
isoperimetric-type inequalities [
6,
7]. Finally, we demonstrate the applications of our results in the temperature research on a planet, and we obtain an
approximate mean temperature formula.
2 Basic concepts and main result
In [
2‐
5,
8], the authors systematically studied the theory of satellite and obtained some results which have the application value. But in space science, the centered surround system
\(S^{(2)} \{ P,\varGamma \}\) [
2‐
5] has its special properties, that is, where the
Γ is an ellipse and
P is one of the foci of the ellipse [
8]. Therefore, it is necessary for us to do further research on this centered surround system.
Let the particle
\(A\in\mathbb{R}^{2}\) be regarded as the Earth, and let its motion trajectory be the ellipse
$$\varGamma \triangleq\biggl\{ x\mathbf{i}+y\mathbf{j}\in\mathbb{R}^{2}\Big| \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, x,y\in\mathbb{R}, a\geq b>0 \biggr\} , $$
where
\(\mathbf{i}=(1,0)\),
\(\mathbf{j}=(0,1)\),
\(\mathbb{R}\triangleq (-\infty,\infty)\),
\(\mathbb{R}^{2}\triangleq\mathbb{R} \times\mathbb{R}\), and let the particle
\(P\triangleq-\sqrt{a^{2}-b^{2}}\mathbf{i}\) be regarded as the Sun, which is a
focus of the ellipse
Γ. Then the set
\(S^{(2)} \{P,\varGamma \}\triangleq \{P,\varGamma \}\) is a centered surround system [
2‐
5].
We remark here that the
foci of the ellipse
Γ are
\(-c \mathbf{i}\) and
c
i, where
\(c=\sqrt{a^{2}-b^{2}}\geq0\), and the
eccentricity of the ellipse
Γ is
\(\mathrm {e}\triangleq{c}/{a} \in[0,1)\). Note that the e in this paper is the eccentricity of the ellipse
Γ rather than the Euler constant
e, that is,
$$\mathrm{e}\ne e\triangleq\lim_{n\rightarrow\infty} \biggl(1+\frac{1}{n} \biggr)^{n}=2.718281828459045\ldots. $$
Let the masses of the Earth
A and the Sun
P be
\(m>0\) and
\(M>0\), respectively. Then, according to the law of universal gravitation, the gravity of the Earth
A to the Sun
P is
$$ \mathbf{F} (A,P )=\frac{GmM (A-P )}{\Vert {{A}-{P}}\Vert ^{3}}, $$
(1)
where the
G is the gravitational constant in the solar system. Without loss of generality, here we assume that
\(GmM=1\).
We say that
$$\bigl\Vert \mathbf{F} (A,P )\bigr\Vert =\frac{1}{\|A-P\|^{2}} \quad \text{and} \quad\overline{\bigl\Vert {\mathbf{F}} ( A,P )\bigr\Vert }\triangleq \frac{1}{|\varGamma |} \oint_{\varGamma }\frac{1}{\|A-P\|^{2}} $$
are the
gravity norm and the
mean gravity norm of the gravity
\(\mathbf{F} (A,P )\), respectively [
5,
9,
10].
In general, the mean gravity norm
\(\overline{\Vert {\mathbf{F}} ( A,P )\Vert }\) cannot be expressed by the elementary functions since it involves the elliptic integral [
11]. Therefore, in order to facilitate the applications, we need to find its sharp lower and upper bounds, which can be expressed by the elementary functions.
In this paper, our main result is as follows.
In [
5], the authors obtained the following
isoperimetric inequality [
6,
7] (see (66) in [
5]):
$$ \overline{\bigl\Vert {\mathbf{F}} ( A,P )\bigr\Vert }\geq\biggl( \frac{2\pi}{|\varGamma |} \biggr)^{2}, $$
(3)
where the
Γ is a smooth and convex Jordan closed curve in
\(\mathbb{R}^{2}\) [
2‐
5,
12]. Obviously, inequalities (
2) are both an improvement and an expansion of inequality (
3).
In space science, the orbit
Γ of a satellite
A is approximate to a circle [
8], that is, the eccentricity e of the ellipse
Γ is very small. Since the error
$$\frac{16-\pi}{4\pi}-\frac{5}{2\pi}=0.22746482927568604\ldots $$
of the coefficients
\({(16-\pi)}/{(4\pi)}\) and
\({5}/{(2\pi)}\) is not very large and the eccentricity e is very small, we see that inequalities (
2) are sharp.
We remark here that, by the isoperimetric inequalities [
5]
$$\pi ab=\operatorname{Area}D(\varGamma )\leq\frac{|\varGamma |^{2}}{4\pi} $$
and the
p-mean inequality [
1,
5,
9,
13‐
16]
$$\begin{aligned} \frac{|\varGamma |}{2\pi} =&\frac{1}{2\pi} \int_{0}^{2\pi} \sqrt{a^{2}\sin^{2} \theta+b^{2}\cos^{2}\theta}\,\mathrm{d}\theta \\ \leq&\sqrt{\frac{1}{2\pi} \int_{0}^{2\pi}{ \bigl(a^{2} \sin^{2}\theta+b^{2}\cos^{2}\theta\bigr)}\, \mathrm{d}\theta} \\ =&\sqrt{\frac{a^{2}+b^{2}}{2}}, \end{aligned}$$
we have
$$ \sqrt{ab}\leq\frac{|\varGamma |}{2\pi}\leq\sqrt{\frac{a^{2}+b^{2}}{2}}. $$
(4)
Therefore, the
\({|\varGamma |}/{(2\pi)}\) is a mean of the positive real numbers
a and
b.
3 Preliminaries
In order to prove Theorem
1, we need to establish several identities and inequalities as follows.
According to the theory of mathematical analysis, we have Lemma
1.
According to the theory of mathematical analysis, we have Lemma
5.
According to the theory of mathematical analysis, we have Lemma
8.
5 Mean temperature on a planet
Let
\(S^{(2)} \{P,\varGamma \}\) be a centered surround system, where
Γ is an ellipse and
P is one of the foci of the ellipse. Then we may think that of
\(A\in \varGamma \) as a planet and of
P as the Sun, and of the ellipse
Γ as the motion trajectory of the planet. Assume that the radiation energy of the Sun
P to the planet
A is 1, then, according to the optical laws, the radiant energy received by the planet
A is
$$C\bigl\Vert \mathbf{F} (A,P )\bigr\Vert =\frac{C}{\|A-P\|^{2}}, $$
where
\(C>0\) is a constant of the radiation energy, and we can measure this constant
C by means of the physical methods.
As everyone knows, the radiant energy
\(C\|\mathbf{F} (A,P )\|\) is important to us. Since the rain and the air humidity are related to the radiation energy, we might think that there exist two constants
\(C_{*}, C^{*}>0\), such that
\(C_{*}\overline{\Vert {\mathbf{F}} ( A,P )\Vert }\) is the
mean air humidity and
\(C^{*}\overline{\Vert {\mathbf{F}} ( A,P )\Vert }\) is the
mean temperature on the Earth in a year. Therefore, Theorem
1 can be used to estimate the mean air humidity and the mean temperature on the Earth in a year.
Suppose that the planet
A is regarded as a particle, and the temperature on the planet
A at a certain moment is
\(T=T(A)\), and the mean temperature on the planet is
T̅. Then, based on the above analysis, there exists a constant
\(C^{*}>0\) such that
$$ T:\varGamma \rightarrow(0,\infty),\qquad T(A)={C^{*}}\bigl\Vert { \mathbf{F}} ( A,P )\bigr\Vert , $$
(51)
and we can measure the constant
\(C^{*}\) by means of some tests. Without loss of generality, here we assume that
\(C^{*}=1\). Then, by (
51), we have
$$ \overline{T}=\overline{\bigl\Vert {\mathbf{F}} ( A,P )\bigr\Vert }. $$
(52)
We remark here that, if the above planet is regarded as a sphere, then the point A will be regarded as the center of the sphere, and the \(T=T(A)\) will be regarded as the maximum temperature on the planet at the moment. In addition, T can also be regarded as the mean of the air humidity on the Earth at a certain moment.
According to Theorem
1 and (
52), we have
$$ \biggl(1+\frac{5}{2\pi}\frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{2}\leq\overline{T} \leq\biggl(1+\frac{16-\pi}{4\pi} \frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi }{|\varGamma |} \biggr)^{2}. $$
(53)
By (
53), we see that there exists a real function
\(\tau(\mathrm {e})\) such that
$$ \overline{T}=\overline{\bigl\Vert {\mathbf{F}} ( A,P )\bigr\Vert }= \biggl(1+\tau(\mathrm{e})\frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{2}, $$
(54)
where
$$ 0.7957747154594768\ldots=\frac{5}{2\pi} \leq\tau(\mathrm{e}) \leq \frac{16-\pi}{4\pi}=1.0232395447351628\ldots. $$
(55)
Hence
$$ \tau(\mathrm{e})\approx\frac{1}{2} \biggl(\frac{5}{2\pi}+ \frac{16-\pi}{4\pi} \biggr)=\frac{26-\pi}{8\pi}=0.9095071300973198\ldots ,\quad \forall\mathrm{e}\in(0,1). $$
(56)
According to (
54) and (
56), we get the following
approximate mean temperature formula:
$$ \overline{T}\approx\biggl(1+0.9095071300973198\ldots\times\frac{\mathrm {e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{2}, \quad\forall\mathrm{e} \in(0,1). $$
(57)
According to (
57), we know that if the eccentricity
\(\mathrm {e}\in(0,1)\) is very small, then the mean temperature
T̅ on the planet is also small. Conversely, if the eccentricity
\(\mathrm{e}\in (0,1)\) is very large, then the mean temperature
T̅ on the planet is also very large. In particular, we have
$$ \lim_{\mathrm{e}\rightarrow0}\overline{T}= \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{2} \quad\text{and}\quad\lim_{\mathrm{e}\rightarrow1} \overline{T}=\infty. $$
(58)
This is the significance of Theorem
1 in the temperature research.
Suppose that the ellipses
Γ and
\(\varGamma _{*}\) are the motion trajectories of two planets, and the Sun
P is one of the foci of the two ellipses; e and
\(\mathrm{e}_{*}\) are the eccentricities of the two ellipses, respectively. Then, by inequality (
53), we have
$$ \biggl(1+\frac{5}{2\pi}\frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}} \biggr) \biggl(\frac{2\pi}{|\varGamma |} \biggr)^{2}> \biggl(1+\frac{16-\pi}{4\pi}\frac{\mathrm {e}_{*}^{2}}{1-\mathrm{e}_{*}^{2}} \biggr) \biggl(\frac{2\pi}{|\varGamma _{*}|} \biggr)^{2}\quad\Rightarrow\quad \overline{T}>\overline{T^{*}}, $$
(59)
where
T̅ and
\(\overline{T^{*}}\) are the mean temperatures on the two planets, respectively. This is also the significance of Theorem
1 in the temperature research.
6 Conclusions
In this paper, we establish the gravity inequalities in the centered surround system
\(S^{(2)} \{P,\varGamma \}\), which are both an improvement and an expansion of inequality (
3), where the
Γ is an ellipse and
P is one of the foci of the ellipse. We also demonstrate the applications of the inequalities in the temperature research on a planet, and we obtain an approximate mean temperature formula; we illustrate the significance of the formula in the temperature research on a planet.
The theoretical significance of this paper is to establish the geometric and physics theories on satellite motion, and the application value is to estimate the mean air humidity and the mean temperature on the Earth in a year. Large pieces of analysis, geometry, physics, and inequality theories are used in this paper, especially the mathematical analysis, and the series [
17] is the crucial one.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.