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Physics & Astronomy International Journal

Review Article Volume 2 Issue 5

Secular influence of time variation of the gravitational constant on the periods of pulsars

Lin Sen Li

School of Physics, Northeast Normal University, Changchun, China

Correspondence: Lin?Sen Li, School of Physics, Northeast Normal University, Changchun, China

Received: May 02, 2018 | Published: October 22, 2018

Citation: Lin–Sen L. Secular influence of time variation of the gravitational constant on the periods of pulsars. Phys Astron Int J. 2018;2(5):488-491. DOI: 10.15406/paij.2018.02.00129

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Abstract

The theoretical formulae for the influence of the change of the theoretical formulae for the influence of the change of moment of inertia due to a time variation of the gravitational constant on the period of a pulsar are given by the method for solving the first order linear differential equations. The analytical and numerical solutions of the period of a pulsar slow down due to time variation of the gravitational constant are derived and calculated for five pulsars (PSR1749–28, PSR2045–16. JP1933+16, HP1508+55 and CP0834+06). Numerical results are given in Table 1 the results are discussed and conclusions are given

Keywords: pulsars–time variation of g–period–influence

Introduction

Some authors studied the variation of pulse period arising from the change of moment of inertia and they always use the method of the angular momentum conservation (L=IΩ=consta ) or energy conservation (Erot=12IΏ2=const ). However, the angular momentum is not conservative due to energy loss arising from the radiating power. Hence the change of the period of pulsar cannot be researched by using the angular momentum and the rotating energy conservation. In the formula of the magnetic dipole model of pulsars the moment of inertia is not variable (constant). But when we consider the energy–loss or time variation of the gravitational constant the moment of inertia may be changed. Hence it is necessary to give the formula for the magnetic dipole model which suits the change of moment of inertia. This is an important work in this paper.

It is well known that the gravitational constant is variable with time since the large number hypothesis suggested by Dirac. However, the variation of gravitational constant influences the change of moment of inertia according to the formula ˙I/I=ε˙G/G . Therefore the variation of G with time also influences the change of the rotational angular velocity or period of pulsar through the change of moment of inertia. Heintzman and Hillebrant1 studied the relation between pulsar slow down and the temporal change of the gravitational constant. They estimated the variable value of gravitational constant˙G/G per year from the above relation. However, they did not research the influence of time variation of G on the change of period of a pulsar. Li2 studied the retardation of rotation of the Earth due to the variation of the gravitational constant, but he did not research spin down of pulsar due to the gravitational constant. In the present paper the author researched pulsar slow down due to time variation of the gravitational constant through the change of moment of inertia

The equation determining the influence of change of moment of inertia on the period of a pulsar

The pulsar radiating power W is transformed from the rotational energy at a rate dEdt , i e

W=dEdt or W+dEdt=0   (1)

According to the theory of magnetic model

dEdt=23c3(Msinα)2Ω4=323c3π4μ2P4,=W. Ω=2πP.   (2)

Hereis the period of pulsar, and μ is the projection of the magnetic dipole moment on the direction perpendicular to the rotational axis. We assume that when we consider time variation of the gravitational constant,μ=μ0 (const), which does not influence the magnetic dipole moment.

The energy carried away by radiation from the rotational energy of pulsar can be written

E=12IΩ2=2π2IP2. (3)

Here I denotes moment of inertia. If we consider the variation of moment of inertia with time, then

˙E=2π2[1P2dIdt2IP3dPdt].   (4)

Substituting the formula (2) and (4) into the formula (1), we obtain the Bernoulli equation for n=1

dPdt12(1IdIdt)P=8π2μ023c3I(t)(P1)   (5)

When both sides of the equation (5) are multiplied by 2P,

2PdPdt˙IIP2=16π2μ203c3I(t)

Or

dP2dt˙IIP2=16π2μ023c3I(t).  

We can transform Bernoulli equation into the first order linear differential equation. i. e. the equation (5) may be written as the form of the first order linear differential equation

dP2dt+NP2=Q(t),  (6)

Comparing the equation (6) with the above equation, we define that

N=˙I/I ,Q(t)=16π2μ023c3I(t)      (7)

The solution of the equation determining influence of time variation of g on the period of pulsars

Some authors give the relation between the variation of moment of inertia I and time variation of the gravitational constant G as follows ( Blake 1978, Will 1981 )3,4

˙I/I=κ˙G/G.   (8)

Blake gives that the coefficient κ lies the range 0.1 to 0.2, and Will4 gives ε=0.17. However, the formula (8) is derived from the equation of hydrostatic equilibrium, which is suitable to an Earth model and does not suite to the neutron stellar model.

Heintzmann & Hillebrant1 studied pulsar slow down and the temporal change of G. They gave the formulas for the connection of the change of moment of inertia with time variation of the gravitational constant for the white dwarf star and neutron star. For the neutron stars

dlnGdlnI=43γAσ2   (9)

According to (7) this can be written as

˙II=(243γAσ)˙GG=N   (10)

Here σ=2GMc2R , M and R denote mass and radius of the neutron star. The parameter A is determined by γ=n+1/n . n is the index of polytropic model. For γ=5/3, 2 ,3 , A=10, 4, 1.˙GG=1013/yr~1012/yr

N=˙II=(243γAσ)˙GG=const.  (11)

According to the first linear differential equation (6), N is a function of timeor it is a constant value. In this paper N is a constant value as shown in the expressions (11) and (20). Integrating (7), yields I=I0eNt , (12) Substitution of (12) into the second expression of (7), we get

Q(t)=16π2μ023c2I0eNt   (13)

Integrating the equation (6), we get

P(t)2=eNdt[Q(t)eNdtdt+C] . C is an integrating constant, Substituting (13) into the above integral expression, we obtain

P(t)2=eNt[C+(16π2μ023c3I0)e2Ntdt].   

When we take t =0,P2(t)=P2(0), C=P (0) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacqGH0icxcaWGdbGaeyypa0JaamiuaiaacIcacaaIWaGaaiyk aSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaaaa@429B@ C=P (0) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacqGH0icxcaWGdbGaeyypa0JaamiuaiaacIcacaaIWaGaaiyk aSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaaaa@429B@ , i.e.

Integrating the above expression, one yields

P (t) 2 = e Ntt [P (0) 2 +( 16 π 2 μ 0 2 3 c 3 I 0 ) 0 t e 2Nt dt. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGqbGaaiikaiaadshacaGGPaGcdaahaaqcbasabKazba4= baqcLbmacaaIYaaaaKqzGeGaeyypa0JaamyzaSWaaWbaaKazba4=be qaaKqzadGaeyOeI0IaamOtaiaadshacaWG0baaaKqzGeGaai4waiaa dcfacaGGOaGaaGimaiaacMcalmaaCaaajqwaa+FabeaajugWaiaaik daaaqcLbsacqGHRaWkcaGGOaGcdaWcaaqaaKqzGeGaaGymaiaaiAda cqaHapaCkmaaCaaajeaibeqcKfaG=haajugWaiaaikdaaaqcLbsacq aH8oqBkmaaBaaajqwaa+FaaKqzadGaaGimaaqcKfaG=hqaaOWaaWba aKazba4=beqaaKqzadGaaGOmaaaaaOqaaKqzGeGaaG4maiaadogakm aaCaaajeaibeqcKfaG=haajugWaiaaiodaaaqcLbsacaWGjbGcdaWg aaqcKfaG=haajugWaiaaicdaaKazba4=beaaaaqcLbsacaGGPaGcda WdXbqaaKqzGeGaamyzaOWaaWbaaKqaGeqabaqcLbmacaaIYaGaamOt aiaadshaaaqcLbsacaGGKbGaaiiDaiaac6caaKqaGeaajugWaiaaic daaKqaGeaajugWaiaadshaaKqzGeGaey4kIipaaaa@88D2@    (14)

In the formula (5) when t=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWG0bGaeyypa0JaaGimaaaa@3D1F@ , I= I 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaqaaaaaaaaaWdbiaadMeacqGH9aqpcaWGjbWcpaWaaSbaaKqa GeaajugWaiaaicdaaKqaGeqaaaaa@3F9F@ , d I 0 dt =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba WaaSaaaeaajugibiaadsgacaWGjbGcdaWgaaqcbasaaKqzadGaaGim aaWcbeaaaOqaaKqzGeGaamizaiaadshaaaGaeyypa0JaaGimaaaa@42B0@ , P=P(0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGqbGaeyypa0JaamiuaiaacIcacaaIWaGaaiykaaaa@3F29@ , P ˙ = P ˙ (0), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaceWGqbGbaiaacqGH9aqpceWGqbGbaiaacaGGOaGaaGimaiaa cMcacaGGSaaaaa@3FEB@ one yields

8 π 2 μ 0 2 3 c 3 I 0 =P(0) P ˙ (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba WaaSaaaeaajugibiaaiIdacqaHapaCkmaaCaaaleqajeaibaqcLbma caaIYaaaaKqzGeGaeqiVd02cdaWgaaqcbasaaKqzadGaaGimaaqcba sabaWcdaahaaqcbasabeaajugWaiaaikdaaaaakeaajugibiaaioda caWGJbWcdaahaaqcbasabeaajugWaiaaiodaaaqcLbsacaWGjbWcda WgaaqcbasaaKqzadGaaGimaaqcbasabaaaaKqzGeGaeyypa0Jaamiu aiaacIcacaaIWaGaaiykaiqadcfagaGaaiaacIcacaaIWaGaaiykaa aa@55E1@    (15)

Substituting the expression (15) into the formula (14), then, the formula (14) become as

P (t) 2 = e kt [P (0) 2 +( P(0) P ˙ (0) N )( e 2Nt 1)] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGqbGaaiikaiaadshacaGGPaWcdaahaaqcbasabeaajugW aiaaikdaaaqcLbsacqGH9aqpcaWGLbWcdaahaaqcbasabeaajugWai abgkHiTiaadUgacaWG0baaaKqzGeGaai4waiaadcfacaGGOaGaaGim aiaacMcakmaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaS IaaiikaOWaaSaaaeaajugibiaadcfacaGGOaGaaGimaiaacMcaceWG qbGbaiaacaGGOaGaaGimaiaacMcaaOqaaKqzGeGaamOtaaaacaGGPa GaaiikaiaadwgalmaaCaaajeaibeqaaKqzadGaaGOmaiaad6eacaWG 0baaaKqzGeGaeyOeI0IaaGymaiaacMcacaGGDbaaaa@6169@    (16)

Hence, we can estimate the variable rate of the pulse period per century as follows

δP=[P(t)P(0)](s/century). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacqaH0oazcaWGqbGaeyypa0Jaai4waiaadcfacaGGOaGaamiD aiaacMcacqGHsislcaWGqbGaaiikaiaaicdacaGGPaGaaiyxaiaacI cacaWGZbGaai4laiaadogacaWGLbGaamOBaiaadshacaWG1bGaamOC aiaadMhacaGGPaGaaiOlaaaa@5105@    (17)

Where P(0) is the initial value as t=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWG0bGaeyypa0JaaGimaaaa@3D1F@ .

Numerical results

We use the formulas (16)—(17) to estimate the periodic variation of five pulsars PSR0843+06, PSR1508+55, PSR1933+16, PSR1749–28 and PSR2045–16 due to time variation of the gravitational constant per century. The P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGqbaaaa@3B3B@ and P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGqbaaaa@3B3B@ of these pulsars are adopted from data in a Table given by Allen5, We assume these pulsars have M=1.4 (solar mass) and R=1.2km,6 the polytropic index n=1 ( Alan, Riper,,1975) σ=2GM/ c 2 R=0.3439 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaqaaaaaaaaaWdbiabeo8aZjabg2da9iaaikdacaWGhbGaamyt aiaac+cacaWGJbWcpaWaaWbaaKqaGeqabaqcLbmapeGaaGOmaaaaju gibiaadkfacqGH9aqpcaaIWaGaaiOlaiaaiodacaaI0aGaaG4maiaa iMdaaaa@4A77@ .(Heintzmann & Hillerant, 1975),1 σ=2GM/ c 2 R=0.3439 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaqaaaaaaaaaWdbiabeo8aZjabg2da9iaaikdacaWGhbGaamyt aiaac+cacaWGJbWcpaWaaWbaaKqaGeqabaqcLbmapeGaaGOmaaaaju gibiaadkfacqGH9aqpcaaIWaGaaiOlaiaaiodacaaI0aGaaG4maiaa iMdaaaa@4A77@ , 43γAσ=3.3756 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaqaaaaaaaaaWdbiaaisdacaGGtaIaaG4maiabeo7aNjaacobi caWGbbGaeq4WdmNaeyypa0Jaai4eGiaaiodacaGGUaGaaG4maiaaiE dacaaI1aGaaGOnaaaa@47C8@ . Substituting these data into the expression (10) which can be written as

N= I ˙ I ={ 2 43γAσ ] G ˙ G =[ 2 3.3756 ] G ˙ G =0.592487 G ˙ G. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGobGaeyypa0JaeyOeI0IcdaWcaaqaaKqzGeGabmysayaa caaakeaajugibiaadMeaaaGaeyypa0JaeyOeI0Iaai4EaOWaaSaaae aajugibiaaikdaaOqaaKqzGeGaaGinaiabgkHiTiaaiodacqaHZoWz cqGHsislcaWGbbGaeq4Wdmhaaiaac2fakmaalaaabaqcLbsaceWGhb GbaiaaaOqaaKqzGeGaam4raaaacqGH9aqpcqGHsislcaGGBbGcdaWc aaqaaKqzGeGaaGOmaaGcbaqcLbsacaaIZaGaaiOlaiaaiodacaaI3a GaaGynaiaaiAdaaaGaaiyxaOWaaSaaaeaajugibiqadEeagaGaaaGc baqcLbsacaWGhbaaaiabg2da9iabgkHiTiaaicdacaGGUaGaaGynai aaiMdacaaIYaGaaGinaiaaiIdacaaI3aGcdaWcaaqaaKqzGeGabm4r ayaacaaakeaajugibiaadEeacaGGUaaaaaaa@68BF@    (18)

We cited the time variation of G ˙ G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba WaaSaaaeaajugibiqadEeagaGaaaGcbaqcLbsacaWGhbaaaaaa@3CB0@ given by AI–Rawaf7

2.8× 10 13 /yr G ˙ G 6.0× 10 13 /yr. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacqGHsislcaaIYaGaaiOlaiaaiIdacqGHxdaTcaaIXaGaaGim aSWaaWbaaKqaGeqabaqcLbmacqGHsislcaaIXaGaaG4maaaajugibi aac+cacaWG5bGaamOCaiablQNiWPWaaSaaaeaajugibiqadEeagaGa aaGcbaqcLbsacaWGhbaaaiablQNiWjabgkHiTiaaiAdacaGGUaGaaG imaiabgEna0kaaigdacaaIWaWcdaahaaqcbasabeaajugWaiabgkHi TiaaigdacaaIZaaaaKqzGeGaai4laiaadMhacaWGYbGaaiOlaaaa@5CC5@    (19)

In this paper we calculate the lower limit for G ˙ G =2.8× 10 13 /yr. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba WaaSaaaeaajugibiqadEeagaGaaaGcbaqcLbsacaWGhbaaaiabg2da 9iabgkHiTiaaikdacaGGUaGaaGioaiabgEna0kaaigdacaaIWaWcda ahaaqcbasabeaajugWaiabgkHiTiaaigdacaaIZaaaaKqzGeGaai4l aiaadMhacaWGYbGaaiOlaaaa@4C32@

Substituting this value for G ˙ G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba WaaSaaaeaajugibiqadEeagaGaaaGcbaqcLbsacaWGhbaaaaaa@3CB0@ into the expression (18), we obtain

N=1.66× 10 13 /yr=constant MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGobGaeyypa0JaaGymaiaac6cacaaI2aGaaGOnaiabgEna 0kaaigdacaaIWaWcdaahaaqcbasabeaajugWaiabgkHiTiaaigdaca aIZaaaaKqzGeGaai4laiaadMhacaWGYbaeaaaaaaaaa8qacqGH9aqp caWGJbGaam4Baiaad6gacaWGZbGaamiDaiaadggacaWGUbGaamiDaa aa@5291@  (20)

Substituting the values of N, P(0), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGqbGaaiikaiaaicdacaGGPaGaaiilaaaa@3DFE@ and P ˙ (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaceWGqbGbaiaacaGGOaGaaGimaiaacMcaaaa@3D57@ into the formula (16) and takes t=100 year MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWG0baeaaaaaaaaa8qacqGH9aqpcaaIXaGaaGimaiaaicda caqGGaGaamyEaiaadwgacaWGHbGaamOCaaaa@431C@ (1st century), we obtain the numerical results listed in Table 1 for the slow down of periods of five pulsars due to a time variation of the gravitational constant.

Table 1 shows that pulse periods of five pulsars are prolonged in the range 0.0000149s~0.0000363s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaqaaaaaaaaaWdbiaaicdacaGGUaGaaGimaiaaicdacaaIWaGa aGimaiaaigdacaaI0aGaaGyoaiaadohacaGG+bGaaGimaiaac6caca aIWaGaaGimaiaaicdacaaIWaGaaG4maiaaiAdacaaIZaGaam4Caaaa @4A96@ per century due to time variation of the gravitational constant

Pulsars

P( t 0 )(s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba acemqcLbsacaWFqbGaa8hkaiaa=rhalmaaBaaajeaibaqcLbmacaWF Waaajeaibeaajugibiaa=LcacaWFOaGaa83Caiaa=Lcaaaa@42C3@

P ˙ ( t 0 )×1 0 -15 (s/s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba acemqcLbsaceWFqbGbaiaacaWFOaGaa8hDaOWaaSbaaKqaGeaajugW aiaa=bdaaSqabaqcLbsacaWFPaGaa831aiaa=fdacaWFWaWcdaahaa qcbasabeaajugWaiaa=1cacaWFXaGaa8xnaaaajugibiaa=HcacaWF ZbGaa83laiaa=nhacaWFPaaaaa@4B35@

P(t)(s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba acemqcLbsacaWFqbGaa8hkaiaa=rhacaWFPaGaa8hkaiaa=nhacaWF Paaaaa@3FD5@

δt(s/cent) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba acemqcLbsacaWF0oGaa8hDaiaa=HcacaWFZbGaa83laiaa=ngacaWF LbGaa8NBaiaa=rhacaWFPaaaaa@4347@

PSR1749–28

0.5625532

 8.15

0.5625789

0.0000257

PSR2045–16

1.9615669

 10.96

1.9616032

0.0000363

JP 1933+16

0.3587354

 6.00

0.3587543

0.0000189

HP 1508+55

0.7396779

 5.04

0.7396928

0.0000149

CP 0834+06

1.2737635

 6.80

1.2737849

0.0000214

Table 1 Numerical results for spin down of periods of five pulsars due to time variation of gravitational constant in the lower limit

Discussions and conclusion

  1. In the quadrupole elastic energy model of neutron stars the total energy and moment of inertia connect with oblateness ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacqaH1oqzaaa@3C0D@ ,8 i e,
  2. E= E 0 + 1 2 I Ω 2 +A ε 2 +B (ε ε 0 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGfbGaeyypa0JaamyraOWaaSbaaKqaGeaajugWaiaaicda aSqabaqcLbsacqGHRaWkkmaalaaabaqcLbsacaaIXaaakeaajugibi aaikdaaaGaamysaiabfM6axTWaaWbaaKqaGeqabaqcLbmacaaIYaaa aKqzGeGaey4kaSIaamyqaiabew7aLLqbaoaaCaaajeaqbeqaaKqzGd GaaGOmaaaajugibiabgUcaRiaadkeacaGGOaGaeqyTduMaeyOeI0Ia eqyTdu2cdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcLbsacaGGPa qcfa4aaWbaaKqaafqabaqcLboacaaIYaaaaaaa@5CB8@ , I= I 0 (1+ε) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGjbGaeyypa0JaamysaSWaaSbaaKqaGeaajugWaiaaicda aKqaGeqaaKqzGeGaaiikaiaaigdacqGHRaWkcqaH1oqzcaGGPaaaaa@449C@ .

    But in the magnetic dipole model we may not consider oblateness ε=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacqaH1oqzcqGH9aqpcaaIWaaaaa@3DCD@ , the ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacqaH1oqzlmaaBaaajeaibaqcLbmacaaIWaaajeaibeaaaaa@3E75@ is the primary value. We consider pulsars as spherical stars.

  3. Some pulsar, such as Crab and Vela speed up suddenly due to stellar quakes at some time.8 They do not effect all pulsars. It is a temporary happning and is not a secular happing. It cannot influence the secular variation of a slow down due to time variation of gravitational constant.
  4. Because the value for μ 0 2 I 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba WaaSaaaeaajugibiabeY7aTTWaaSbaaKqaGeaajugWaiaaicdaaKqa GeqaaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaaGcbaqcLbsacaWGjb WcdaWgaaqcbasaaKqzadGaaGimaaqcbasabaaaaaaa@44A4@ cannot be obtained from the observation, it may be written as the formula (14) in terms of P(0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGqbGaaiikaiaaicdacaGGPaaaaa@3D4E@ and P ˙ (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsaceWGqbGbaiaacaGGOaGaaGimaiaacMcaaaa@3D57@
  5. Hence, the formula (14) can be expressed by using the formula (16)

  6. In this paper the results are obtained under the condition of no variation effect for the magnetic dipole moment and magnetic inclination without variation.

We also obtained the conclusions:

  1. The variation of the gravitational constant with time may be determined known from the observation and theories. It connects with the large number hypothesis in cosmology
  2. Time variation of gravitational constant can influences the change of moment of inertia through.
  3. The change of moment of inertia can influences the spin down of the period of pulsar due to time variation of the gravitational constant, and the variation of the moment of inertia is an exponential formulation under the condition for time variation of the gravitational constant.
  4. The variable rate of the spin down of the period of five pulsars are on the order 10–5 seconds per century. This effect can be observed by the current astronomical instruments over a long time.9

Acknowledgements

None.

Conflict of interest

The author declares no conflict of interest.

References

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