• Chinese Physics B
  • Vol. 29, Issue 10, (2020)
Moxian Qian, Xibin Li, and Yongjun Cao
Author Affiliations
  • College of physics and electronic information, Inner Mongolia Normal University, Hohhot 010022, China
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    DOI: 10.1088/1674-1056/ab9f24 Cite this Article
    Moxian Qian, Xibin Li, Yongjun Cao. Gravitation induced shrinkage of Mercury’s orbit[J]. Chinese Physics B, 2020, 29(10): Copy Citation Text show less

    Abstract

    In general relativity, Mercury’s orbit becomes approximately elliptical and the its perihelion has thus an additional advance. We demonstrate, meanwhile, that in comparison of those given by Newton’s theory of gravitation for the orbit of the Mercury, the circumference and the area are reduced by 40.39 km and 2.35 × 109 km2, respectively, besides the major-axis contraction pointed out recently, and all are produced by the curved space within Einstein's theory of gravitation. Since the resolution power of present astronomical distance measurement technology reaches one kilometer, the shrinkage of Mercury’s orbit can then be observable.
    R(θ)=rgsb211+ecosθ,(1)

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    ds2=(12rgsr)dt2+dr212rgs/r+r2(dθ2+sin2θdϕ2),(2)

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    rrgs[1b2(1+ecost)2(3+2e2%+e2cost)(1+ecost)2sin2(t2)]=R(t)rgsf(t),(3)

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    f(t)=2(3+2e2+e2cost)(1+ecost)2sin2(t2)(4)

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    t(θ)(13b2)θ,orθ(1+3b2)t(θ).(5)

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    Δθ3b2Δt(θ)=6πb2.(6)

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    d=dx2+dy2=dR2+(Rdθ)2=R2+(dRdθ)2dθ.(7)

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    n=02πR2+(dRdθ)2dθ.(8)

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    Sn=1202πR(θ)d=1202πR(θ)(dRdθ)2+R2dθ.(9)

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    d2=112rgs/rdr2+r2dθ2,(10)

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    d=112rgs/r(drdθ)2+r2dθ,(11)

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    =02π+6πb2112rgs/r(drdθ)2+r2dθ=02π112rgs/r(drdt)2+r2dt.(12)

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    r2R(t)22rgsR(t)f(t),(13a)

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    drdtdR(t)dtrgsdf(t)dt,(13b)

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    (drdt)2(dR(t)dt)22rgsdR(t)dtdf(t)dt.(13c)

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    112rgs/r(drdt)2(dR(t)dt)22rgsdR(t)dtdf(t)dt+2rgs1R(t)(dR(t)dt)2.(14)

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    112rgs/r(drdt)2+r2(dR(t)dt)2+R(t)22rgsQ(t),(15)

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    Q(t)=dR(t)dtdf(t)dt+R(t)f(t)1R(t)(dR(t)dt)2.(16)

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    112rgs/r(drdt)2+r2(dRdt)2+R2rgsQ(t)(dRdt)2+R2.(17)

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    02π112rgs/r(drdt)2+r2dt02π(dRdt)2+R2dtrgs02πQ(t)(dRdt)2+R2dtnrgs02πQ(t)(dRdt)2+R2dt.(18)

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    02πQ(t)(dRdt)2+R2dt27.48.(19)

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    Δ=rgs02πQ(t)(dRdt)2+R2dt27.48rgs40.39km.(20)

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    dS=12rd.(21)

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    S=1202π+6πb2rd(θ)=1202πrd(t).(22)

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    rd(t)dt=r112rgs/r(drdt)2+r2(R(t)rgsf(t))×((ddt)2+R2rgsQ(t)(dRdt)2+R2)=R(t)(dRdt)2+R2rgs×(R(t)Q(t)(dRdt)2+R2+f(t)(dRdt)2+R2).(23)

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    S1202πR(t)(dRdt)2+R2dtrgs1202π(R(t)Q(t)(dRdt)2+R2+f(t)(dRdt)2+R2)dtSnrgsG,(24)

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    G=1202π(R(t)Q(t)(dRdt)2+R2+f(t)(dRdt)2+R2)dt,(25)

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    G28.78rgsb2.(26)

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    ΔS=28.78rgs2b2=2.35×109km2.(27)

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    Moxian Qian, Xibin Li, Yongjun Cao. Gravitation induced shrinkage of Mercury’s orbit[J]. Chinese Physics B, 2020, 29(10):
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