R E S E A R C H
Open Access
Gravity inequalities and the mean
temperature on a planet
Jia Jin Wen
1, Jun Yuan
2*, Shan He Wu
3and Tian Yong Han
1*Correspondence: yuanjun_math@126.com 2School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, Jiangsu 211171, P.R. China Full list of author information is available at the end of the article
Abstract
In the centered surround systemS(2){P,
Γ
}, we establish the following gravityinequalities:
1 + 5
2
π
e2 1 –e2
2
π
|
Γ
| 2≤ 1
|
Γ
|Γ
1 A–P2 ≤
1 +16 –
π
4π
e2 1 –e2
2
π
|
Γ
| 2,
where
Γ
is an ellipse,Pandeare one of the foci and the eccentricity of the ellipse, respectively, andA∈Γ
is a satellite of the centered surround systemS(2){P,Γ
}. 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 –e2
2
π
|
Γ
| 2, ∀e∈(0, 1),
where theTis the mean temperature on a planet.
MSC: 26D15; 26E60; 51K05; 52A40
Keywords: centered surround system; mean gravity norm; isoperimetric inequality; Newton formula; Euler formula
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 themean temperatureon 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 []. To this end, we first introduce the basic concepts in the surround system [–], 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 systemS(){P,Γ}, in particular, themean gravity norm formula, as well as we
il-lustrate the coefficients in these inequalities are the best constants. Next, we provegravity inequalitiesin the centered surround systemS(){P,Γ}, which are alsoisoperimetric-type
inequalities[, ]. Finally, we demonstrate the applications of our results in the tempera-ture research on a planet, and we obtain anapproximate mean temperature formula. 2 Basic concepts and main result
In [–, ], the authors systematically studied the theory of satellite and obtained some results which have the application value. But in space science, the centered surround sys-temS(){P,Γ}[–] has its special properties, that is, where theΓ is an ellipse andPis
one of the foci of the ellipse []. Therefore, it is necessary for us to do further research on this centered surround system.
Let the particleA∈R be regarded as the Earth, and let its motion trajectory be the
ellipse
Γ
xi+yj∈Rx
a+
y
b = ,x,y∈R,a≥b>
,
where i = (, ), j = (, ),R(–∞,∞),RR×R, and let the particleP–√a–bi
be regarded as the Sun, which is afocusof the ellipseΓ. Then the setS(){P,Γ}{P,Γ}
is a centered surround system [–].
We remark here that thefociof the ellipseΓ are –ciandci, wherec=√a–b≥,
and theeccentricityof the ellipseΓ is ec/a∈[, ). Note that the e in this paper is the eccentricity of the ellipseΓ rather than the Euler constante, that is,
e=e lim n→∞
+
n
n
= . . . . .
Let the masses of the EarthAand the SunPbem> andM> , respectively. Then, according to the law of universal gravitation, the gravity of the EarthAto the SunPis
F(A,P) = GmM(A–P)
A–P , ()
where theGis the gravitational constant in the solar system. Without loss of generality, here we assume thatGmM= .
We say that
F(A,P)=
A–P and F(A,P)
|Γ|
Γ
A–P
are the gravity norm and the mean gravity norm of the gravity F(A,P), respectively [, , ].
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.
Theorem (Gravity inequalities) Let S(){P,Γ} be a centered surround system, where
theΓ is an ellipse and the P is one of the foci of the ellipse.Then we have the following isoperimetric-type inequalities[, ]:
+ π
e
– e
π |Γ|
≤F(A,P)≤
+ –π π
e
– e
π |Γ|
, ()
where theeis the eccentricity of the ellipse.The equalities in()hold if and only ifΓ is a circle.
In [], the authors obtained the followingisoperimetric inequality[, ] (see () in []):
F(A,P)≥
π |Γ|
, ()
where theΓ is a smooth and convex Jordan closed curve inR[–, ]. Obviously,
in-equalities () are both an improvement and an expansion of inequality ().
In space science, the orbitΓ of a satelliteAis approximate to a circle [], that is, the eccentricity e of the ellipseΓ is very small. Since the error
–π π –
π = . . . .
of the coefficients ( –π)/(π) and /(π) is not very large and the eccentricity e is very small, we see that inequalities () are sharp.
We remark here that, by the isoperimetric inequalities []
πab=AreaD(Γ)≤|Γ|
π
and thep-mean inequality [, , , –] |Γ|
π = π
π
asinθ+bcosθdθ
≤
π
π
asinθ+bcosθdθ
=
a+b
,
we have
√
ab≤|Γ|
π ≤
a+b
. ()
3 Preliminaries
In order to prove Theorem , we need to establish several identities and inequalities as follows.
According to the theory of mathematical analysis, we have Lemma .
Lemma Let f :R→Rbe a periodic function with the period T.Then we have
T+t
t
f(x) dx=
T
f(x) dx, ∀t∈R. ()
Lemma For any continuous function g: [, ]→R,we have
π
π
gcosθdθ= π
π
gcosθdθ. ()
Proof Since the functiong(cosθ) is both an even function and a periodic function with
the periodπ, according to Lemma , we have
π
π
gcosθdθ = π
π
–π
gcosθdθ
= π
π
gcosθdθ
= π
π
–π
gcosθdθ
= π
π
gcosθdθ.
This ends the proof of Lemma .
Lemma (Mean gravity norm formula) Under the hypotheses in Theorem,we have
F(A,P)=
π |Γ|
π
π
√
– ecosθdθ
π
π
+ ecosθ ( – ecosθ)/dθ
. ()
Proof SinceΓ ={xi+yj∈R|x=acosθ,y=bsinθ,θ∈[, π]}, we have A–P=(acosθ+c)+ (bsinθ)
=
acosθ+ accosθ+c+a–csinθ
=a+ccosθ=a( + ecosθ), that is,
A–P=a( + ecosθ). ()
Since
ds
dθ =
dx
dθ
+
dy
dθ
by () and (), we have
F(A,P)= |Γ|
Γ
A–Pds
= |Γ|
π
a( + ecosθ)
dx
dθ
+
dy
dθ
dθ
= |Γ|
π
a√ – ecosθ
a( + ecosθ) dθ
=
a|Γ|
π
√
– ecosθ
( + ecosθ) dθ
=
a|Γ|
π
√
– ecosθ ( + ecosθ)/dθ,
that is,
F(A,P)=
a|Γ|
π
√
– ecosθ
( + ecosθ)/dθ. ()
In (), setθ =ϑ–π. Since the cosine functioncosxis a periodic function with the period π, by Lemma , we have
F(A,P)=
a|Γ|
π
√
– ecosθ ( + ecosθ)/dθ
=
a|Γ|
π+π
π
√
– ecos(ϑ–π)
[ + ecos(ϑ–π)]/d(ϑ–π)
=
a|Γ|
π+π
π
√
+ ecosϑ ( – ecosϑ)/dϑ
=
a|Γ|
π
√
+ ecosϑ ( – ecosϑ)/dϑ
=
a|Γ|
π
√
+ ecosθ ( – ecosθ)/dθ,
that is,
F(A,P)=
a|Γ|
π
√
+ ecosθ
( – ecosθ)/dθ. ()
By () and (), we get
F(A,P)=
a|Γ|
π
√
– ecosθ ( + ecosθ)/ +
√
+ ecosθ ( – ecosθ)/
dθ
=
a|Γ|
π
( – ecosθ)+ ( + ecosθ)
( – ecosθ)/ dθ
=
a|Γ|
π
that is,
F(A,P)=
a|Γ|
π
+ ecosθ
( – ecosθ)/dθ. ()
On the other hand, by (), we have
|Γ|=
π
dx
dθ
+
dy
dθ
dθ=a
π
√
– ecosθdθ. ()
According to (), (), and Lemma , we get
F(A,P)=
a|Γ|
π
+ ecosθ
( – ecosθ)/dθ
= |Γ|
π
√
– ecosθdθ π
+ ecosθ ( – ecosθ)/dθ
=
π |Γ|
π
π
√
– ecosθdθ
π
π
+ ecosθ
( – ecosθ)/dθ
.
That is, equation () is proved. The proof of Lemma is completed.
Lemma The inequality
π
π
√
– ecosθdθ≥ –δe ()
holds if and only if
δ≥ –
π, ()
where theδis a positive constant.
Proof Assume that () holds, we prove that () holds as follows. By ≤e < , ≤ecosθ< and the Newton formula
( +x)μ= +
∞
n=
n!
n
k=
(μ+ –k)xn, ∀x∈(–, ), ()
we get
√
– ecosθ= + ∞
n=
n!
n
k=
+ –k –e
cosθn
= – e
∞
n=
(n– )!! (n– )(n)!!e
(n–)cosnθ,
that is,
√
– ecosθ= – e ∞
n=
(n– )!! (n– )(n)!!e
where
(n)!!××× · · · ×(n) and (n– )!!××× · · · ×(n– ). ()
By (), we get
π
π
√
– ecosθdθ= – e ∞
n=
(n– )!! (n– )(n)!!e
(n–)
π
π
cosnθdθ . ()
Since
π
π
√
–cosθdθ=
π
π
sinθdθ=
π, ()
in (), letting e→, we get
∞
n=
(n– )!! (n– )(n)!!
π
π
cosnθdθ = –
π. ()
From (), (), (), and ≤e < , we get
π
π
√
– ecosθdθ = – e ∞
n=
(n– )!! (n– )(n)!!e
(n–)
π
π
cosnθdθ
≥ – e
∞
n=
(n– )!! (n– )(n)!!
π
π
cosnθdθ
= –
–
π e
≥ –δe.
That is, inequality () is proved.
Next, assume that () holds; we prove () as follows. In (), letting e→, by (), we get
π ≥ –δ ⇔ δ≥ –
π.
That is, inequality () holds. This proves Lemma .
According to the theory of mathematical analysis, we have Lemma .
Lemma For any positive integer n,we have
π
π
cosnθdθ=(n– )!!
(n)!! , ()
Lemma The inequality
π
π
√
– ecosθdθ≤√ –εe ()
holds if and only if
ε≤
, ()
where theεis a positive constant.
Proof Assume that () holds, we prove () holds as follows.
By thep-mean inequality [, , , –] and Lemma , we see that, for any positive integern, we have
π
π
cosnθdθ≥
π
π
cosθdθ
n
=
n. ()
In (), use the following exchanges
e↔√e
and θ↔,
we get
– e
∞
n=
(n– )!! (n– )(n)!!e
(n–)
n =
– e
. ()
From (), (), (), and (), we get
π
π
√
– ecosθdθ
= – e
∞
n=
(n– )!! (n– )(n)!!e
(n–)
π
π
cosnθdθ
≤ – e
∞
n=
(n– )!! (n– )(n)!!e
(n–)
n
=
– e
≤√ –εe.
Hence inequality () is proved.
Next, assume that () holds; we prove () as follows. Let < e < . By (), we get
ε≤ e
–
π
π
√
– ecosθdθ
By () and Lemma , we get
ε≤lim e→
e – π π √
– ecosθdθ
=lim t→
t – π π √
–tcosθdθ
=lim t→
d dt – π π √
–tcosθdθ
= –lim t→ π π √
–tcosθdθ d
dt π π √
–tcosθdθ
= –lim t→
d dt π π √
–tcosθdθ
= –lim t→ π π d dt √
–tcosθdθ
= –lim t→ π π
–cosθ
√ –tcosθdθ
= π
π
cosθdθ
= .
Hence inequality () holds. The proof of Lemma is completed.
Lemma For any positive integer n,we have
≤
n+ n+
n
k=
– k <
π. ()
Proof Recall the following famous Euler formula:
sinx=x
∞
n=
– x
πn , ∀x∈R. ()
In (), setx=π/, we get
lim n→∞ n k= – k =
∞
n=
– n =
π. ()
Define an auxiliary sequence{un}∞n=as follows:
{un}∞n=: un
n+ n+
n
k=
–
k . ()
By (), we have
lim n→∞un=
Since
un+
un
=n+ n+ ×
n+ n+
– (n+ )
=n
+ n+ n+
n+ n+ n+ > ,
the sequence{un}∞n=is strictly increasing. Therefore, by (), we have
=u≤un<nlim→∞un=
π.
That is, inequalities () holds. This ends the proof of Lemma .
According to the theory of mathematical analysis, we have Lemma .
Lemma Let the sequence{un}∞n=⊂Rbe convergent,and letlimn→∞un=μ.Then we
have
lim N→∞
N
N
n=
un=μ. ()
Lemma The inequality
π
π
+ ecosθ
( – ecosθ)/dθ≤ +μ
e
– e ()
holds if and only if
μ≤
π, ()
where theμis a positive constant.
Proof Assume that () holds, we prove () as follows. By the Newton formula (), we get
+ ecosθ
( – ecosθ)/
= – ecosθ–/– – ecosθ–/
= +
∞
n=
n!
n
k=
–
+ –k –e
cosθn
–
∞
n=
n!
n
k=
–
+ –k –e
cosθn
= + e
∞
n=
(n+ )!! (n)!! e
(n–)cosnθ
– e
∞
n=
(n– )!! (n)!! e
= + e
∞
n=
(n+ )!! – (n– )!! (n)!! e
(n–)cosnθ
= + e
∞
n=
(n+ )(n– )!! (n)!! e
(n–)cosnθ,
that is,
+ ecosθ
( – ecosθ)/ = + e
∞
n=
(n+ )(n– )!! (n)!! e
(n–)cosnθ. ()
Since
n
k=
–
k = (n+ )
(n– )!! (n)!!
, ()
by (), (), and Lemma , we get
π
π
+ ecosθ
( – ecosθ)/dθ
= + e
∞
n=
(n+ )(n– )!! (n)!! e
(n–)
π
π
cosnθdθ
= + e
∞
n=
(n+ )(n– )!! (n)!! e
(n–)(n– )!!
(n)!!
= + e ∞
n=
n+ n+
n
k=
– k e
(n–),
that is,
π
π
+ ecosθ
( – ecosθ)/dθ= + e
∞
n=
n+ n+
n
k=
– k e
(n–). ()
By (), ≤e < , and Lemma , we get
π
π
+ ecosθ
( – ecosθ)/dθ
= + e
∞
n=
n+ n+
n
k=
– k e
(n–)
≤ + πe
∞
n=
e(n–)
= + π
e – e
≤ +μ e
– e.
Hence inequality () is proved.
Let < e < . According to (), we see that inequality () can be rewritten as
+ e
∞
n=
n+ n+
n
k=
– k e
(n–)≤ +μe ∞
n=
e(n–).
Hence
μ≥
∞ n=n+n+
n
k=( –k)e(n–)
∞ n=e(n–)
= limN→∞
N n=n+n+
n
k=( –k)e(n–)
limN→∞ N
n=e(n–)
= lim N→∞
N n=n+n+
n
k=( –k)e(n–)
N n=e(n–)
= lim N→∞
N
n=une(n–) N
n=e(n–)
,
that is,
μ≥ lim N→∞
N
n=une(n–) N
n=e(n–)
, ∀e∈(, ), ()
where the sequence{un}∞n=is defined by ().
In (), letting e→, by (), (), and Lemma , we get
μ≥lim e→Nlim→∞
N
n=une(n–) N
n=e(n–)
= lim N→∞
N
N
n=
un=
π.
This proves inequality (). The proof of Lemma is completed.
Lemma The inequality
π
π
+ ecosθ
( – ecosθ)/dθ≥ +ν
e
– e ()
holds if and only if
ν≤
, ()
where theνis a positive constant.
Proof Assume that () holds, we prove () as follows. By (), ≤e < , and Lemma , we have
π
π
+ ecosθ
( – ecosθ)/dθ
= + e
∞
n=
n+ n+
n
k=
– k e
≥ + e
∞
n=
e(n–)
= +
e
– e
≥ +ν e
– e.
Hence inequality () is proved.
Next, assume that () holds, we prove () as follows.
Let < e < . According to (), we see that inequality () can be rewritten as
+ e
∞
n=
n+ n+
n
k=
– k e
(n–)≥ +νe ∞
n=
e(n–).
Hence
ν≤
∞
n=une(n–) ∞
n=e(n–)
, ∀e∈(, ), ()
where the sequence{un}∞n=is defined by ().
In (), letting e→, we get
ν≤u=
.
Hence inequality () is proved. This ends the proof of Lemma .
Lemma The inequality
–
–
π e
+
e
– e ≥ +η
e
– e ()
holds if and only if
η≤
π, ()
where theηis a positive constant.
Proof Let < e < . Then inequality () can be rewritten as
η≤ – e
e
–
–
π e
+
e
– e –
= π +
–
–
π e
.
Hence inequality () holds if and only if
η≤ inf <e<
π +
–
–
π e
= π +
–
–
π =
π.
Lemma The inequality
–e
+ π
e
– e ≤ +τ
e
– e ()
holds if and only if
τ≥ –π
π , ()
whereτ is a positive constant.
Proof Let
< e < , ϕ(e)– + (
π – )e
(
–e + ) +
π.
Then inequality () can be rewritten as
τ≤ – e
e
–e
+ π
e
– e –
= – + (
π – )e
(
–e+ ) +
π =ϕ(e).
Since <π< , the functions
+
π – e
and
–e
+
–
are strictly increasing, we see that the functionϕ(e) is strictly decreasing. So we see that inequality () holds if and only if the inequality
τ≥ sup <e<
ϕ(e)=ϕ() = –π π
holds. That is, () holds if and only if () holds. This ends the proof of Lemma .
Remark According to Lemmas and , we see that
–
–
π e
≤
π
π
√
– ecosθdθ≤
– e
, ()
where the coefficients – /πand / of eare the best constants. Set
ω(t)
– t–
–
–
π t
, t∈[, ),
then
ω(t) = –
√ – t+ –
π >ω
() = –
√+ – π
Therefore, the error
– e
–
–
–
π e
=ωe<ω() = √ –
π
= . . . .
is very small.
Remark According to Lemmas and , we see that
+
e
– e≤
π
π
+ ecosθ
( – ecosθ)/dθ≤ +
π
e
– e, ()
where the coefficients / and /πof e/( – e) are the best constants, and the error
π –
= . . . .
is also very small.
Remark According to Lemma , we see that
–
–
π e
+
e
– e ≥ +
π
e
– e, ()
where the coefficient /(π) of e/( – e) is the best constant.
Remark According to Lemma , we know that
–e
+ π
e
– e ≤ +
–π π
e
– e, ()
where the coefficient ( –π)/(π) of e/( – e) is also the best constant.
4 Proof of Theorem 1
Proof According to Lemma , and Remarks , , and , we have
F(A,P)=
π |Γ|
π
π
√
– ecosθdθ
π
π
+ ecosθ
( – ecosθ)/dθ
≥
π |Γ|
–
–
π e
+
e
– e
≥
π |Γ|
+ π
e
– e .
According to Lemma , and Remarks , , and , we have
F(A,P)=
π |Γ|
π
π
√
– ecosθdθ
π
π
+ ecosθ
( – ecosθ)/dθ
≤
π |Γ|
– e
+ π
e
– e
≤
π |Γ|
+ –π π
e
– e .
Hence the second inequalities in () also hold.
Based on the above analysis, we see that equalities in () hold if and only if e = , that is, Γ is a circle. This completes the proof of Theorem .
5 Mean temperature on a planet
LetS(){P,Γ}be a centered surround system, whereΓ is an ellipse andPis one of the foci
of the ellipse. Then we may think that ofA∈Γ as a planet and ofPas the Sun, and of the ellipseΓ as the motion trajectory of the planet. Assume that the radiation energy of the SunPto the planetAis , then, according to the optical laws, the radiant energy received by the planetAis
CF(A,P)= C A–P,
whereC> is a constant of the radiation energy, and we can measure this constantCby means of the physical methods.
As everyone knows, the radiant energyCF(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 constantsC∗,C∗> , such thatC∗F(A,P)is themean air humidityandC∗F(A,P)is themean temperatureon the Earth in a year. Therefore, Theorem can be used to estimate the mean air humidity and the mean temperature on the Earth in a year.
Suppose that the planetAis regarded as a particle, and the temperature on the planet
Aat a certain moment isT=T(A), and the mean temperature on the planet isT. Then, based on the above analysis, there exists a constantC∗> such that
T:Γ →(,∞), T(A) =C∗F(A,P), ()
and we can measure the constantC∗by means of some tests. Without loss of generality, here we assume thatC∗= . Then, by (), we have
T=F(A,P). ()
According to Theorem and (), we have
+ π
e – e
π |Γ|
≤T≤
+ –π π
e – e
π |Γ|
. ()
By (), we see that there exists a real functionτ(e) such that
T=F(A,P)=
+τ(e) e
– e
π |Γ|
, ()
where
. . . . =
π ≤τ(e)≤ –π
π = . . . . . ()
Hence
τ(e)≈
π +
–π π =
–π
π = . . . . , ∀e∈(, ). ()
According to () and (), we get the followingapproximate mean temperature for-mula:
T≈
+ . . . .× e
– e
π |Γ|
, ∀e∈(, ). ()
According to (), we know that if the eccentricity e∈(, ) is very small, then the mean temperatureTon the planet is also small. Conversely, if the eccentricity e∈(, ) is very large, then the mean temperatureTon the planet is also very large. In particular, we have
lim e→T=
π |Γ|
and lim
e→T=∞. ()
This is the significance of Theorem in the temperature research.
Suppose that the ellipsesΓ andΓ∗ are the motion trajectories of two planets, and the
SunPis one of the foci of the two ellipses; e and e∗are the eccentricities of the two ellipses,
respectively. Then, by inequality (), we have
+ π
e
– e
π |Γ|
>
+ –π π
e ∗
– e ∗
π |Γ∗|
⇒ T>T∗, ()
whereTandT∗are the mean temperatures on the two planets, respectively. This is also the significance of Theorem in the temperature research.
6 Conclusions
In this paper, we establish the gravity inequalities in the centered surround system
S(){P,Γ}, which are both an improvement and an expansion of inequality (), where the
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 theo-ries 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 [] is the crucial one.
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.
Author details
1Institute of Mathematical Inequalities and Applications, Chengdu University, Chengdu, Sichuan 610106, P.R. China. 2School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, Jiangsu 211171, P.R. China.3Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, P.R. China.
Acknowledgements
The authors would like to acknowledge the support from the National Natural Science Foundation of China
(No. 11161024 and No. 61309015), and the Natural Science Foundation of Fujian Province of China (No. 2016J01023), and the Natural Science Foundation of Sichuan Province Technology Department (No. 2014SZ0107).
Received: 12 June 2016 Accepted: 4 October 2016 References
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