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The second order perturbation effect of gravitational radiation damping on the periastron advance of binary stars is studied. The second order analytic solution is obtained based on the first order theory in the 2014 article by Li. Theoretical results show that secular variation exists in the periastron advance of binary stars in the second order theory, but secular variation does not exist in the first order perturbation theory. Numerical results for two compact binary stars (PSR J0737-3039 and M33 X-7) are given, demonstrating the theoretical significance even though the effect is very small.
It is important and significant to study the secular effect of gravitational radiation damping on the time and space variation of the periastron of binary stars. The time variation refers to the time variation of periastron passage. The space variation of periastron refers to the advance of periastron. Lincoln & Will (1990) deduced from the equation for time variation of periastron passage that there is no secular variation of longitude of periastron ω, but only relativistic periastron advance in 5/2 post-Newtonian (PN) (gravitational-radiation emission) of first order perturbation theory. Li (2011) obtained the secular and periodic solutions of the time variation of periastron passage for binary stars. Moreover, Li (2009) studied gravitational radiation damping and the orbital evolution of compact binary pulsars by using the first perturbation method presented in Walker & Will (1979). In that paper, the author only obtained the periodic variation of longitude of periastron and not secular variation, that is, there is no periastron advance in first order perturbation theory for 5/2 PN. Li (2014) also studied gravitational radiation damping and the orbital evolution of compact binary stars by using the first perturbation method presented in Lincoln & Will (1990). However, in that paper the author obtained periodic variation and cases without secular variation of the longitude of periastron, that is, there is no periastron advance in first order perturbation theory for 5/2 PN.
It is necessary to examine the existence of secular variation of the orbit or periastron advance of binary stars in second order perturbation theory. The associated results demonstrate the existence of secular variation in the orbital elements of a binary star system in second order perturbation theory.
2 A Method for Solving the Perturbation Equations in High Order Perturbation Theory
It is necessary to research and explore how high order perturbation theory is related to the stability of a binary star system or the solar system. The most important consideration is the secular variation of orbital elements in high order perturbation theory. Some authors investigate this topic, such as the book Brouwer & Clemence (1961).
Perturbation equations describing time as an independent variable may be written as
where σ denotes one of the orbital elements, a is the semi-major and e is the eccentricity.
Equation (1) can be transformed into an equation with true anomaly f as the independent variable from the case of time being an independent variable
The perturbation order of the orbital elements may be written as
Comparing Equation (4) with Equation (5), we find that
The purpose of this paper is to investigate the secular effect of second order perturbation on the longitude of periastron, i.e. letting σ = ω
Integrating Equation (11), we yield
where ω is the argument of periastron and ϖ is the longitude of periastron.
3 Results for the First Order Perturbation Effect of Gravitational Radiation Damping on the Orbit of Binary Stars
Results for the first order perturbation effects have been given by Li (2014) and related results are listed as follows.
The formula for relative acceleration with 5/2 PN was provided by Lincoln & Will (1990)
For gravitational emission, we take
where m = m_{1} + m_{2} and r denotes the distance between the two binary stars. n and V denote the unit vectors of the radial direction and the relative velocity vector respectively.
Resolving the perturbation acceleration a into the radial component , the transverse component perpendicular to in the orbital plane and the component normal to the orbital plane, we obtain (Li 2014)
Here and G = c = 1. These parameters have been defined in Lincoln & Will (1990). denotes the true anomaly, and a and e denote the semi-major axis and eccentricity respectively.
We substitute , and from Equations (16)–(18) into Gaussian equations (16)–(22) in Li (2014), which were derived from Lincoln & Will (1990) and Brouwer & Clemence (1961). We also change independent variable time t to an independent true anomaly f to transform the Gaussian equations by using
. Then, by integrating Gaussian equations (16)–(22), we obtain the results for the first order perturbation effects in 5/2 PN as follows (Li 2014).
The coefficients of the secular terms are
The amplitudes of the periodic terms are
where .
Expressions (19)–(30) indicate that there are both secular and periodic variations for the semi-major axis and the eccentricity, but there is only periodic variation for the argument of periastron ω in first order perturbation theory; that is, there is neither secular variation of the longitude of periastron, ϖ nor periastron advance for 5/2 PN in first order perturbation theory.
4 Secular Solution for Longitude of Periastron in Second Order Theory
Because there is no secular variation (shift) for the longitude of periastron in the first order theory, it is necessary to examine the secular variation (shift) for the longitude of periastron in second order theory.
First, one must write down the first order Gaussian equations for the longitude of argument ω with the anomaly as an independent variable by using Gaussian equations (Lincoln & Will 1990)
By substituting Equations (16)–(18) for , and respectively into the above Gaussian equation and using , the first order Gaussian equation with true anomaly as an independent variable for ω has been derived by Li (2014)
For brevity, we write down the first order equation above in the following form
where
The second order equation for the longitude of periastron can be written according to Equation (7).
where
Here
where .
The expressions in Equation (19) with (36) yield
We take the secular term
Similarly the expressions in Equation (20) with Equation (38) give
In addition, Equations (21)–(24) with Equation (40) generate
Substituting Equations (43), (44) and (45) into Equation (35), one obtains
Integrating the above equation from 0 to 2π
We use the formula for integration
Similarly
The result of the integration (47) is
where P is the orbital period.
where ϖ and ω are the longitudes of periastron and argument respectively.
5 Numerical Results for the Secular Effect of Gravitational Radiation Damping on Periastron Advance in Second Order Perturbation Theory
In this paper we choose two compact binary star systems. One is PSR J0737–3039 and the other is black hole binary star M33 X-7. Their data are listed in Table 1.
Table 1
Table for data on PSR J0737–3039 and M33 X-7
The right hand side of Equations (16) – (18) needs to be multiplied by c^{5} (light speed) and m should be multiplied by G (Gravitational constant).
Here m_{1}, m_{2} and a_{0} are expressed in the units of solar mass , g, and solar radius . cm, (cgs) and .
Substituting data for the values of Equations (53)–(54) and M_{1} (), M_{2} () and e_{0} into Equations (27), (28), (37) and (39), we obtain Tables 2 and 3.
Amplitude
i = 1
−23.8290
−1.3401
+47.5940
−9.1810
i = 2
−1.7468
−0.3855
+3.4493
−1.2651
i = 3
−0.1986
−0.0227
+0.0485
−0.8924
i = 4
0
0
+0.0003
−0.0628
Table 2
Numerical values for the amplitudes of periodic terms for PSR J0373-3039
Amplitude
i = 1
–1.7178
–1.4119
+73621
–231.7082
i = 2
–0.0146
–4.1082
+33.1253
–31.8699
i = 3
–0.0002
–0.0056
+4.9418
–44.4767
i = 4
0
0
+0.0003
–0.7462
Table 3
Numerical Values for the Amplitudes of Periodic Terms for M33 X-7
Expanding the terms in Equations (50) and (51) for
The next expression is similar to the above expression
Substituting Equations (55) and (56) into Equations (50) and (51), we obtain that
We can substitute the values of A_{1}, A_{3}, A_{4}, Q_{1}, Q_{3}, Q_{4}, E_{1}, E_{3}, E_{4}, N_{1}, N_{3} and N_{4} into Table 2, Table 3 and P (d) in Table 1.
Then, we can apply these results for PSR J0737-3039 and M33 X-7 in Equations (57) and (58) to obtain the second order secular solutions for periastron advance of pulsars or black holes that are part of binary systems as shown in Table 4.
Binary Star
(rad cycle^{−1})
(rad yr^{−1})
(rad cycle^{−1})
(rad yr^{−1})
PSR J0737–3039
0
0
6.55 × 10^{−24}
2.34 × 10^{−20}
M33 X–7
0
0
3.97 × 10^{−23}
4.21 × 10^{−21}
Notes: The first order values of and have been given in a previous article (Li 2014).
Table 4The second order secular solutions for the periastron advance of PSR J0737–3–039 and M33 X–7 compared with the first order solution of the previous paper (Li 2014).
6 DISCUSSION
References
BrouwerD.ClemenceG. M.1961
BurgayM.D'AmicoN.PossentiA.et al.2003
BurgayM.D'AmicoN.PossentiA.et al.2005Astronomical Society of the Pacific Conference Series328
LiL.-S.2009
LiL.-S.2011
LiL.-S.2014
LincolnC. W.WillC. M.1990
MatukumaT.HirayamaK.1930
OroszJ. A.McClintockJ. E.NarayanR.et al.2007
WalkerM.WillC. M.1979
WillemsB.KalogeraV.HenningerM.2004
Cite this article:
Li Lin-Sen. The secular effect of gravitational radiation damping on the periastron advance of binary stars in second order perturbation theory. Cancer Biol Med. 2017; 8:084.