_{1}

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Perturbation methods are employed to calculate time variation in the orbital elements of a compact binary system. It turns out that the semi-major axis and eccentricity exhibit only periodic variations. The longitude of periastron and mean longitude of epoch exhibit both secular and periodic variation. In addition, the relativistic effects on the time of periastron passage of binary stars are also given. Four compact binary systems (PSRJ0737-3039, PSR1913+16, PSR1543+12 and M33X-7) are considered. Numerical results for both secular and periodic effects are presented, and the possibility of observing them is discussed.

In the wake of unceasing development in the post-Newtonian celestial mechanics, at present, the research on the post-Newtonian effects has been exhibited gradually due to the fact that the degree of accuracy of astronomical observation improves unceasingly. Hence several authors devoted their research to this subject and the scopes (Brumberg, (1972, 1985) [

The relative acceleration of two-body with the Post-Newtonian Parameters is given by Will (1981) [^{ }

a P N = m X r 3 [ ( 2 γ + 2 β ) m r − γ v 2 + ( 2 + α 1 − 2 ζ 2 ) μ r − 1 2 ( 6 + α 1 + α 2 + α 3 ) μ m v 2 + 3 2 ( 1 + α 2 ) μ m ( v ⋅ n ) 2 ] + m ( X ⋅ v ) v ¯ r 3 [ ( 2 γ + 2 ) − μ m ( 2 − α 1 + α 2 ) ] . (1)

here m = m 1 + m 2 , μ = m 1 m 2 m , n = X r , X = n ⋅ r = r , r ⋅ t ˙ = r ⋅ r ˙ , v = r ˙ = r ˙ n + r f ˙ λ , v 2 = r ˙ 2 + r 2 f ˙ 2 , ( v ⋅ n ) 2 = ( v ¯ ⋅ X r ) 2 = ( v ⋅ n r ) 2 r 2 = ( r ˙ 2 ⋅ r ) 2 r 2 = r ˙ 2

m ( X ⋅ v ) v r 3 = m r 3 ( n r ⋅ v ) v = m r 3 ( r ⋅ r ˙ ) r ˙ = m r 3 ( r ⋅ r ˙ ) ( r ˙ n + r f ˙ λ ) = m r 3 ( r r ˙ n + r 2 r ˙ f ˙ λ ) . (2)

here f denotes the true anomaly. n is a unit vector in the radial direction and λ are unit vectors in the orbital plane. n is directed along the radial direction, and λ is perpendicular to n . In the equation m denotes Gm and the right side should be multiplied by c^{−2}. G is gravitational constant and c is the speed of light.

In this paper we research the general relativistic effect. In the general relativistic case the Post-Newtonian parameters α 1 = α 2 = α 3 = 0 , β = 1 , γ = 1 , ζ 2 = 0. (Will 1981) [

a P N = m X r 3 [ 4 m r − v 2 + 2 μ r − 3 μ m v 2 + 3 2 μ m ( v ⋅ n ) 2 ] + m ( X ⋅ v ) v ¯ r 3 [ 4 − 2 μ m ] . (3)

Using the relative expressions the Equation (3) may be written

a p p n = m r 3 ( r n ) [ 4 m r + 2 μ r − { 1 + 3 μ m ) ( r ˙ 2 + r 2 f ˙ 2 ) + 3 2 μ m r ˙ 2 ] + m r 3 [ ( r r ˙ 2 n + r 2 r ˙ f ˙ λ ) { 4 − 2 μ m } ] . (4)

here boldface denotes vector.

We resolve the acceleration a into a radial component R n , a component S λ , normal to R n and a component W normal to the orbital plane.

i.e, a = R n + S λ + W ( n × λ ) , n × λ = N (the unit vector normal to the orbital plane).

On comparison with the expression (4), we get three scalar accelerative components R, S and W

R = m r 2 [ 4 m r − 2 μ r − ( 1 + 3 μ m ) ( r ′ 2 + r 2 f ˙ 2 ) + 3 2 r ˙ 2 μ m ] + m r 2 r ˙ 2 [ 4 − 2 μ m ] S = m r [ 4 − 2 μ m ] r ˙ f ˙ W = 0 (5)

Substituting the following formulas of the problem of two body into the above formula (Smart, 1953) [^{ }

f ˙ = d f d t = n a 1 − e 2 2 / r 2 , r ˙ = n a e sin f 1 − e 2 , n 2 a 3 = m . (6)

We obtain

r 2 R = m 2 ( ( 4 + 2 μ m ) r + [ 3 − 7 2 μ m ] e 2 sin 2 f p − ( 1 + 3 μ m ] p r 2 ] , r 2 S = m 2 r [ 4 − 2 μ m ] e sin f , W = 0. (7)

here p = a ( 1 − e 2 ) .

An independent variable dt is transformed to an independent variable df in the Gaussian perturbation equations (Brouwer & Clemence, 1961 [

d a d f = 2 a m ( 1 − e 2 ) [ R r 2 e sin f + S r 2 ( p r ) ] , d e d f = 1 n 2 a 3 [ R r 2 sin f + S r 2 ( cos E + cos f ) ] , d ω ˜ d f = 1 n a e [ − R r 2 cos f + S r 2 ( 1 + r p ) sin f ] , d ε 0 d f = − 2 r 3 R n 2 a 4 1 − e 2 + e 2 1 + 1 − e 2 d ω d f , d i d f = r 3 cos ( ω + f ) n 2 a 4 ( 1 − e 2 ) W , d Ω d f = r 3 sin ( ω + f ) n 2 a 4 ( 1 − e 2 ) W , } (8)

where ω ˜ is the longitude of periastron, E is the eccentric anomaly and ε 0 is the mean longitude at epoch.

Substituting R, S and W for expressions (7) into the perturbation Equation (8) by using Kepler’s third law n 2 a 3 = ( m 1 + m 2 ) = m , we obtain

d a d f = 2 m ( 1 − e 2 ) 2 { [ ( 7 − 3 μ m ) e + ( 3 − 31 μ 8 m ) e 3 ] sin f + ( 5 − 4 μ m ) e 2 sin 2 f − 3 8 μ m e 3 sin 3 f } d e d f = m p { [ ( 3 − μ m ) + ( 7 − 47 8 μ m ) e 2 ] sin f + ( 5 − 4 μ m ) e sin 2 f − 3 8 e 2 μ m sin 3 f } , d ω ˜ d f = m e p { 3 e + [ ( μ m − 3 ) + ( 1 + 21 8 μ m ) e 2 ] cos f − ( 5 − 4 μ m ) e cos 2 f + 3 8 μ m e 2 cos 3 f } , d i d f = d Ω d f = 0 , d ε 0 d f = − 2 m a 1 − e 2 { ( 6 − 9 2 μ m ) + ( 3 − 7 2 μ m ) e 2 − 1 p r − ( 4 − 1 2 μ m ) e cos f } + e 2 1 + 1 − e 2 d ω d f . (9)

Integrating the above Equation (9), we obtain the perturbation secular and periodic solutions

δ a = − 2 m ( 1 − e 2 ) 2 { [ ( 7 − 3 μ m ) e + ( 3 − 31 8 μ m ) e 3 ] ( cos f − cos f 0 ) + 1 2 ( 5 − 4 μ m ) e 2 ( cos 2 f − cos 2 f 0 ) − 1 8 μ m e 3 ( cos 3 f − cos 3 f 0 ) } , δ e = − m a ( 1 − e 2 ) { [ ( 3 − μ m ) + ( 7 − 47 8 μ m ) e 2 ] ( cos f − cos f 0 ) + 1 2 ( 5 − 4 μ m ) e ( cos 2 f − cos 2 f 0 ) − 1 8 e 2 μ m ( cos 3 f − cos 3 f 0 ) } , δ ω ˜ = m a ( 1 − e 2 ) e { 3 e ( f − f 0 ) + [ ( μ m − 3 ) + ( 1 + 21 8 μ m ) e 2 ] ( sin f − sin f 0 ) − 1 2 ( 5 − 4 μ m ) e ( sin 2 f − sin 2 f 0 ) + 1 8 μ m e 2 ( sin 3 f − sin 3 f 0 ) } , δ ε 0 = − m a 1 − e 2 { ( 12 − 9 μ m ) ( f − f 0 ) − ( 6 − 7 μ m ) 1 − e 2 ( E − E 0 ) − ( 8 − μ m ) e ( sin f − sin f 0 ) } + e 2 1 + 1 − e 2 δ ω , δ λ = n ( t − t 0 ) + δ ε 0 , δ i = δ Ω = 0 , m = m 1 + m 2 } (10)

where λ denotes the mean longitude of periastron, E denotes the eccentric anomaly. In the last integral expression, we have used the next integral already:

∫ r d f = a ∫ 1 − e 2 d E ∫ e 2 sin 2 f p r d f = e 2 − 1 p ∫ r d f + ∫ d f − e ∫ cos f d f .

1) The secular variation per cycle (revolution)

By letting f 0 = 0 , f = 2 π , E 0 = 0 , E = 2 π , the periodic terms are disappeared and one obtains the secular variables per cycle (revolution):

Δ a = 0 ( cm / cycle ) Δ e = 0 ( /cycle ) Δ ω ˜ = 6 π m a ( 1 − e 2 ) ( rad / cycle ) Δ ε 0 = − { 2 π m a 1 − e 2 [ ( 12 − 9 μ m ) − ( 6 − 7 μ m ) ( 1 − e 2 ) 1 / 2 ] − 6 π m a ( 1 − e 2 ) ( e 2 1 + 1 − e 2 ) } ( rad / cycle ) Δ λ = ( 2 π + Δ ε 0 ) ( rad / cycle ) Δ i = Δ Ω = 0 ( rad / cycle ) } (11)

where μ = G ( M + m )

The author Li (2010) [

Δ τ = 2 π a 1 / 2 m 1 / 2 [ ( 9 − 2 μ m ) − 1 2 ( 7 − 17 μ m ) e + ( 51 / 2 − 24 μ m ) e 2 ] ( s / Rev ) . (12)

2) The secular variable rate:

a ˙ = e ˙ = i ˙ = Ω ˙ = 0 ω ˜ ˙ = Δ ω ˜ / P ( rad / yr ) ε ˙ 0 = Δ ε 0 / P ( rad / yr ) λ ˙ = [ 2 π / P + Δ ε 0 / P ] ( rad / yr ) τ ˙ = Δ τ / P ( s / d ) } (13)

where the period P is denoted in unit of day

3) The periodic variation of amplitudes:

In the expressions (10) all terms are the periodic variable terms except for all secular terms. Here we list the maximal and minimal amplitudes of the periodic terms for semi-major axis, a and eccentricity, e from the expressions (10).

For the semi-major axis:

A max = − 2 m ( 1 − e 2 ) 2 [ ( 7 − 3 μ m ) e + ( 3 − 31 8 μ m ) e 3 ] ; A min = + m 4 ( 1 − e 2 ) 2 μ m e 3 . } (14)

For eccentricity:

E max = − m a ( 1 − e 2 ) [ ( 3 − μ m ) + ( 7 − 47 8 μ m ) e 2 ] ; E min = + m 8 a ( 1 − e 2 ) μ m e 2 . } (15)

We use the formulae (11) - (13) to calculate the secular of the general relativistic secular effect on the orbital elements of four compact binary systems. It is convenient to reduce the formulas (11) - (13) to practical units m_{1}, m_{2} and a are denoted by the unit in solar mass M 1 ( m ⊙ ) , M 2 ( m ⊙ ) and solar radius A ( R ⊙ ) , and P is denoted by the unit in day = 86400 s, G = 6.67 × 10^{−8}, (c. g. s), c = 3 × 10^{10} cm/s. The formulae (11) - (13) can be written by taking the secular effect

Δ ω ˜ = 6 K ( M 1 + M 2 ) A ( 1 − e 2 ) ( rad / cycle ) Δ ε 0 = − { 2 K ( M 1 + M 2 ) A 1 − e 2 [ ( 12 − 9 μ m ) − ( 6 − 7 μ m ) ( 1 − e 2 ) 1 / 2 ] − 6 K ( M 1 + M 2 ) A ( 1 − e 2 ) ( e 2 1 + 1 − e 2 ) } ( rad / cycle ) Δ λ = ( 2 π + Δ ε 0 ) ( rad / cycle ) Δ i = Δ Ω = 0 ( rad / cycle ) Δ τ = 2 A 1 / 2 ( M 1 + M 2 ) 1 / 2 [ ( 9 − 2 ( μ M + M 2 ) − 1 2 ( 7 − 17 μ M 1 + M 2 ) e + ( 51 / 2 − 24 μ M 1 + M 2 ) e 2 ] ( s / Rev ) . (16)

here K = π G M ⊙ / c 2 R ⊙ = 6.66 × 10 6 (c, g, s).,

This paper chooses four compact binary systems: PSR1913+16, PSR1543+12, PSRJ0737-3039 and a black hole M33 X-7 as an example. For these compact binary stars, their data for P, a, e, M and m are retrieved from Burgay et al. (2003) [

Substituting these data in

Compact binary stars | P(d) | A(R_{⊙}) | e | M_{1}(m_{⊙}) | M_{2}(m_{⊙}) | Ref |
---|---|---|---|---|---|---|

M33 x-7 | 3.450 | 42.40 | 0.0385 | 15.65 | 70.00 | Orosz et al. (2007) [ |

PSR J0737-3039 | 0.1022 | 1.26 | 0.0878 | 1.34 | 1.25 | Willems et al. (2004) [ |

PSR1913+16 | 0.3230 | 2.80 | 0.6170 | 1.44 | 1.38 | Willems et al. (2004) [ |

PSR1543+12 | 0.1022 | 3.28 | 0.2736 | 1.35 | 1.33 | Konacki et al. (2003) [ |

Compact binary stars | Semi-major axis | Eccentricity | ||
---|---|---|---|---|

A max ( km ) | A min ( km ) | E max ( × 10 − 5 ) | E min ( × 10 − 6 ) | |

M33 X-7 | −1661.80 | 0.00027 | −1.23 | 0.00012 |

PSR J0737-3039 | −48.55 | 0.00016 | −1.23 | 0.00100 |

PSR1913+16 | −51.77 | 0.0560 | −1.68 | 0.0410 |

PSR1543+12 | −60.83 | 0.0037 | −0.62 | 0.00270 |

It can be seen from

It can be seen from

It can be seen from

1) The comparison of the theoretical results with the observable results.

The theoretical results in this paper as compared with the observable results given by several authors for three compact binary stars are listed in the

It can be seen from the above

Compact binary stars | Δ a ( cm / Re ) | e/Re | Δ ω ˜ ("/Re) | Δ ε 0 ("/Re) | Δ τ ( s / Re ) |
---|---|---|---|---|---|

M33 X-7 | 0 | 0 | 16".68 | −32".34 | 11.04 |

PSR J0737-3039 | 0 | 0 | 17.08 | −29.92 | 0.37 |

PSR1913+16 | 0 | 0 | 13.45 | −19.75 | 0.78 |

PSR1543+12 | 0 | 0 | 7.29 | −13.48 | 0.53 |

[Note] The symbol denotes arc-second: Rev denotes Revolution (cycle).

Compact binary stars | a ˙ ("/yr) | e ˙ ("/yr) | ω ˜ ˙ | ε ˙ 0 | τ ˙ ( s / yr ) | ||
---|---|---|---|---|---|---|---|

("/yr) | (deg/yr) | ("/yr) | (deg/yr) | ||||

M33 X-7 | 0 | 0 | 1765″ | 0.49 | −3424″ | −0.95 | 19.47 |

PSR J0737-3039 | 0 | 0 | 61016 | 16.95 | −106875 | −29.68 | 18.45 |

PSR1913+16 | 0 | 0 | 15218 | 4.22 | −22332 | −6.20 | 14.70 |

PSR1543+12 | 0 | 0 | 6323 | 1.75 | −11703 | −3.25 | 7.66 |

Compact binary stars | The theoretical values ω ˜ ˙ (deg/yr) | The observable values ω ˜ ˙ (deg/yr) | Authors for providing Data |
---|---|---|---|

PSRJ0737-3039 | 16.95 | 16.90 16.88 | Kramer et al. (2005) [ |

PSR1913+16 | 4.2260 | 4.2266 | Weisberg & Taylor (2005) [ |

PSR1543+12 | 1.7566 | 1.7558 | Kohacki et al. (2003) [ |

2) The possibility of observing effects.

In the solar system the advance of perihelion of Mercury may be observed by the recent instrument. We can see from

We have four conclusions:

a) The compact binary stars are the best objects for studying the post-Newto- nian effects on the orbits

b) Although there are no secular variation for the semi-major axis and eccentricity, there is maximal amplitude of the periodic terms for semi-major axis, such as | A max | = 1661.80 km .

c) The longitudes of periastron and the mean longitude at epoch exist both secular and periodic variable terms, and the maximal values for ω ˙ and ε ˙ 0 arrive at 16.95/yr and −29.68/yr for PSRJ0737-3039 respectively.

d) The longest time of periastron passage is 19.47 minute per year for black hole M33 X-7. This corresponds to over 3 seconds per a day.

Li, L.S. (2017) General Relativistic Orbital Effects in Compact Binary Stars (Solution by the Method of Celestial Mechanics). World Journal of Mechanics, 7, 360-369. https://doi.org/10.4236/wjm.2017.712027