# Spin contribution to the perihelion precession in binary systems like OJ287:
higher order corrections.

Carlos Marín<EMAIL_ADDRESS>Department of Physics, Universidad San
Francisco de Quito, Diego de Robles y Vía Interoceánica  Jorge Poveda
<EMAIL_ADDRESS>Department of Physics, Universidad San Francisco de
Quito, Diego de Robles y Vía Interoceánica

###### Abstract

Higher order corrections are obtained for the perihelion precession in binary
systems like OJ287, Sagittarius A*-S2 and H1821+643 using both the
Schwarzschild metric and the Kerr metric to take into account the spin effect.
The corrections are performed considering the third root of the motion
equation and developing the expansion in terms of $\epsilon\equiv
r_{s}/\left(a(1-e^{2})\right)$ and $\epsilon^{*}\equiv\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\epsilon$.The results are compared with those obtained
in a previous paper.

Perihelium advance and binary systems and orbits

##  I Introduction

In a previous paper MarinPoveda , higher order corrections (up to n-th order)
were obtained for the perihelion precession (see figure (1)) in binary systems
like OJ287, Sagittarius A*-S2 and H1821+643 using the Schwarzschild metric and
complex integration. The corrections were performed considering the third root
of the motion equation and developing the expansion in terms of
$\epsilon\equiv r_{s}/\left(a(1-e^{2})\right)$, where $r_{s}$ is the
Schwarzschild radius.The results were compared with other expansions that
appear in the literature giving corrections to second and third order Fokas ;
Tyler ; Rosales ; Biesel ; DEliseo ; Scharf ; Do_Nhat . In this paper we will
consider both Schwarzschild and Kerr metrics to take into account the spin
effect and developing the expansion in terms of both $\epsilon\equiv
r_{s}/\left(a(1-e^{2})\right)$ and $\epsilon^{*}\equiv\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\epsilon$, where $E^{\prime}$ and $J$ are the energy and
angular momentum per unit mass and $\alpha$ is a factor proportional to the
black hole spin.

Kerr’s black holes are very interesting because most of the black holes in the
universe probably have a rotational movement (spin) and one of the most
important consequences of that spin is that spacetime is dragged around the
rotating black hole leading to an effect that is known as ”frame dragging”. In
the case of a Schwarzschild black hole, a precession occurs which is the
rotation of the elliptical orbit in the fixed plane of that orbit. For Kerr’s
solution, the drag of the reference frame introduces an additional precession
of the plane of the orbit around the axis of rotation of the black hole in the
same direction of rotation of said black hole. To facilitate the calculations
we will assume that the axis of rotation of the Kerr black hole is
perpendicular to the plane of the orbit.

Figure 1: Perihelion precession. $\omega$ is the initial inclination of the
orbit and $\delta\omega$ is the angle of precession.

##  II Spin contribution to the perihelion precession

The Schwarzschild metric describes a body with spherical symmetry, but without
electric charge and rotational movement. If we take into account that many
black holes that have been found have rotation on their own axis, as for
example the binary system OJ287, one might think that this rotation should
influence the advance of the perihelion of the elliptical orbits. The rotation
around its own axis is given by the angular momentum of spin $S_{z}$ of the
massive body M.

To include the spin in the calculation, you can use the Kerr metric. The Kerr
metric is a solution to the field equations in vacuum for a body of mass M
that rotates on its own axis with an angular momentum $S_{z}$. The Kerr metric
is Misner ; Ryder ; Hobson ; Chandrasekhar ; tHooft ; Ludvigsen ; Marin :

$\displaystyle\left(ds\right)^{2}=c^{2}(d\tau)^{2}=\gamma
c^{2}(dt)^{2}-\frac{r^{2}}{\Delta}(dr)^{2}-r^{2}\left(1+\frac{\alpha^{2}}{r^{2}}+\frac{r_{s}\alpha^{2}}{r^{3}}\right)(d\phi)^{2}+\frac{2r_{s}\alpha}{r}cdtd\phi,$
(1)

where $\alpha=\frac{S_{z}}{Mc}$, $\Delta=r^{2}-r_{s}r+\alpha^{2}$, with
coordinates $x^{0}=ct$, $x^{1}=r$, $x^{2}=\theta$ and $x^{3}=\phi$.
$r_{s}=\frac{2GM}{c^{2}}$ is the Schwarzschild radius. We have taken
$\theta=\pi/2$ (equatorial plane). In this paper we only intend to observe how
the spin could contribute to the perihelion precession and if it is relevant
to the calculation, for which only first order terms in $\frac{\alpha}{r}$
will be taken into account. By doing this, the metric is reduced to:

$\displaystyle\left(ds\right)^{2}=c^{2}(d\tau)^{2}=\gamma
c^{2}(dt)^{2}-\frac{1}{\gamma}(dr)^{2}-r^{2}(d\phi)^{2}+\frac{2r_{s}\alpha}{r}cdtd\phi.$
(2)

The arc length $ds$ satisfies the relation
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$, then, the covariant metric tensor is:

$\displaystyle
g_{\mu\nu}=\left(\begin{array}[]{cccc}\gamma&0&0&\frac{r_{s}\alpha}{r}\\\
0&-\gamma^{-1}&0&0\\\ 0&0&0&0\\\
\frac{r_{s}\alpha}{r}&0&0&-r^{2}.\end{array}\right)$ (7)

The geodesic equation can be written in an alternative form using the
Lagrangian

$\displaystyle
L\left(x^{\mu},\frac{dx^{\mu}}{d\sigma}\right)=-g_{\alpha\beta}\left(x^{\mu}\right)\frac{dx^{\alpha}}{d\sigma}\frac{dx^{\beta}}{d\sigma}$
(8)

where $\sigma$ is a parameter of the trajectory of the particle, which is
usually taken to be the proper time, $\tau$ for a massive particle. Using the
Euler-Lagrange equations:

$\displaystyle\frac{\partial L}{\partial
x^{\mu}}-\frac{d}{d\sigma}\left(\frac{\partial
L}{\partial\left(\frac{dx^{\mu}}{d\sigma}\right)}\right)=0$ (9)

we get the geodesic equation for the particle:

$\displaystyle\frac{du_{\mu}}{d\sigma}=\frac{1}{2}\left(\partial_{\mu}g_{\alpha\beta}\right)u^{\alpha}u^{\beta}$
(10)

where $u_{\mu}=\frac{dx_{\mu}}{d\sigma}$.

With $\sigma=\tau$, for the coordinates $ct$ ($\mu=0$) and $\phi$ ($\mu=3$),
the geodesic equation (10) give us, respectively:

$\displaystyle\frac{d}{d\tau}\left(\gamma
c^{2}\left(\frac{dt}{d\tau}\right)+\frac{r_{s}\alpha
c}{r}\left(\frac{d\phi}{d\tau}\right)\right)=0,$ (11)

$\displaystyle\frac{d}{d\tau}\left(r^{2}\left(\frac{d\phi}{d\tau}\right)-\frac{r_{s}\alpha
c}{r}\left(\frac{dt}{d\tau}\right)\right)=0$ (12)

Both of these equations define the following constants along the trajectory of
the particle around the massive object:

$\displaystyle\gamma c^{2}\left(\frac{dt}{d\tau}\right)+\frac{r_{s}\alpha
c}{r}\left(\frac{d\phi}{d\tau}\right)=E^{\prime},$ (13)

$\displaystyle r^{2}\left(\frac{d\phi}{d\tau}\right)-\frac{r_{s}\alpha
c}{r}\left(\frac{dt}{d\tau}\right)=J$ (14)

where $E^{\prime}$ has units of energy per unit mass and $J$ of angular
momentum per unit mass.

From these equations we get:

$\frac{d\phi}{d\tau}=\frac{1}{\gamma
cr^{2}}\left(\frac{r_{s}\alpha}{r}E^{\prime}+\gamma cJ\right)$ (15)

$\frac{dt}{d\tau}=\frac{1}{\gamma
cr^{2}}\left(\frac{r^{2}}{c}E^{\prime}-\frac{r_{s}\alpha}{r}J\right)$ (16)

Equation (2) can be written as:

$c^{2}=\gamma
c^{2}\left(\frac{dt}{d\tau}\right)^{2}-\frac{1}{\gamma}\left(\frac{dr}{d\tau}\right)^{2}-r^{2}\left(\frac{d\phi}{d\tau}\right)^{2}+\frac{2r_{s}\alpha}{r}c\frac{dt}{d\tau}\frac{d\phi}{d\tau},$
(17)

and replacing the values of $\frac{dt}{d\tau}$ and $\frac{d\phi}{d\tau}$:

$\gamma^{2}r^{4}c^{4}=\gamma
c^{2}\left(\frac{r^{2}}{c}E^{\prime}-\frac{r_{s}\alpha}{r}J\right)^{2}-\gamma
c^{2}r^{4}\left(\frac{dr}{d\tau}\right)^{2}-r^{2}\left(\frac{r_{s}\alpha}{r}E^{\prime}+\gamma
cJ\right)^{2}\qquad\qquad$

$\qquad\qquad\qquad\qquad\qquad\qquad+\frac{2r_{s}\alpha}{r}\left(r^{2}E^{\prime}-\frac{r_{s}\alpha}{r}cJ\right)\left(\frac{r_{s}\alpha}{r}E^{\prime}+\gamma
cJ\right).$

Simplifying and taking only first order terms in $\frac{\alpha}{r}$:

$\gamma r^{4}c^{4}=c^{2}\left(\frac{r^{4}}{c^{2}}E^{\prime
2}-\frac{2rr_{s}\alpha}{c}JE^{\prime}\right)-c^{2}r^{4}\left(\frac{dr}{d\tau}\right)^{2}-r^{2}\left(2c\frac{r_{s}\alpha}{r}JE^{\prime}+\gamma
c^{2}J^{2}\right)+\frac{2r_{s}\alpha}{r}\left(r^{2}cJE^{\prime}\right).$

Introducing the value of $\gamma$ in the left side of the last equation we
have:

$c^{2}-\frac{r_{s}c^{2}}{r}=\frac{1}{c^{2}}E^{\prime
2}-\frac{2r_{s}\alpha}{cr^{3}}JE^{\prime}-\left(\frac{dr}{d\tau}\right)^{2}-\gamma\frac{J^{2}}{r^{2}}$

Finally we obtain an energy conservation equation, similar to the one obtained
in the case of the Schwarzschild metric, but with an additional crossed term
proportional to $JE^{\prime}$ :

$\displaystyle\frac{E^{\prime
2}}{c^{2}}-c^{2}=\left(\frac{dr}{d\tau}\right)^{2}+\gamma\frac{J^{2}}{r^{2}}-\frac{r_{s}c^{2}}{r}+\frac{2r_{s}\alpha}{cr^{3}}JE^{\prime}$
(18)

This alllows us to define an effective potential:

$\widetilde{V}=\gamma\frac{J^{2}}{r^{2}}-\frac{r_{s}c^{2}}{r}+\frac{2r_{s}\alpha}{cr^{3}}JE^{\prime}$
(19)

From equation (18), we can get the expression for the radial kinetic energy
per unit mass:

$\displaystyle\left(\frac{dr}{d\tau}\right)^{2}=A+\frac{r_{s}c^{2}}{r}-\frac{J^{2}}{r^{2}}+\frac{J^{2}r_{s}}{r^{3}}-\frac{2r_{s}\alpha}{cr^{3}}JE^{\prime},$
(20)

where $A=\frac{E^{\prime 2}}{c^{2}}-c^{2}$.

Recall now that since the orbit is an ellipse, there are two points in which
the temporal derivative becomes zero, and they are aphelion and perihelion.
For these two points we can write:

$\displaystyle
A+\frac{r_{s}c^{2}}{R_{a}}-\frac{J^{2}}{R_{a}^{2}}+\frac{J^{2}r_{s}}{R_{a}^{3}}-\frac{2r_{s}\alpha}{cR_{a}^{3}}JE^{\prime}=0$
(21)

$\displaystyle
A+\frac{r_{s}c^{2}}{R_{p}}-\frac{J^{2}}{R_{p}^{2}}+\frac{J^{2}r_{s}}{R_{p}^{3}}-\frac{2r_{s}\alpha}{cR_{p}^{3}}JE^{\prime}=0,$
(22)

where $R_{a}=a\left(1+e\right)$ and $R_{p}=a\left(1-e\right)$.

For an ellipse, the equation of motion have three real and positive roots. Two
of the roots are $R_{a}$ and $R_{p}$ and the other we will call
$R^{\prime}_{o}$. To obtain this root ($R^{\prime}_{o}$) we use equation (20)
:

$\left(\frac{dr}{d\tau}\right)^{2}\frac{r^{3}}{J^{2}}=\frac{A}{J^{2}}r^{3}+\frac{r_{s}c^{2}}{J^{2}}r^{2}-r+r_{s}-\frac{2r_{s}\alpha
E^{\prime}}{cJ}=0,$

and write:

$\frac{A}{J^{2}}r^{3}+\frac{r_{s}c^{2}}{J^{2}}r^{2}-r+r_{s}-\frac{2r_{s}\alpha
E^{\prime}}{cJ}=\frac{A}{J^{2}}(r-R_{a})(r-R_{p})(r-R^{\prime}_{o}).$

The last equation can be written as:

$\frac{r_{s}c^{2}}{J^{2}}r^{2}-r+\left(r_{s}-\frac{2r_{s}\alpha}{cJ}E^{\prime}\right)=\frac{\left|A\right|}{J^{2}}\left(R^{\prime}_{o}+R_{a}+R_{p}\right)r^{2}\qquad\qquad\qquad\qquad$

$\qquad\qquad\qquad\qquad\qquad\qquad-\frac{\left|A\right|}{J^{2}}\left(R_{p}R^{\prime}_{o}+R_{a}R_{p}+R_{a}R^{\prime}_{o}\right)r+\frac{\left|A\right|}{J^{2}}R_{a}R_{p}R^{\prime}_{o}.$

It is important to recall that $A$ is negative for elliptic orbits, so it can
be written as $A=-\left|A\right|$.

Comparing the coefficients of $r^{0}$, $r$ and $r^{2}$ of both sides of the
last equation, and replacing the values of $R_{a}$ and $R_{p}$ we obtain the
following relations:

$\displaystyle\frac{r_{s}c^{2}}{J^{2}}=\frac{\left|A\right|}{J^{2}}\left(R^{\prime}_{o}+2a\right),$
(23)

$\displaystyle
1=\frac{\left|A\right|}{J^{2}}\left(2aR^{\prime}_{o}+a^{2}\left(1-e^{2}\right)\right)$
(24)

$\displaystyle
R^{\prime}_{o}=\frac{J^{2}r_{s}}{\left|A\right|\left(1-e^{2}\right)a^{2}}\left(1-\frac{2\alpha}{cJ}E^{\prime}\right)$
(25)

The value of $R^{\prime}_{0}$ given in (25) is the same result that we
obtained in a previous paper MarinPoveda , but with an extra factor
$1-\frac{2\alpha}{cJ}E^{\prime}$. Using equations (24) and (25) we can finally
write:

$\displaystyle
R^{\prime}_{0}=\frac{a\left(1-e^{2}\right)r_{s}\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)}{\left(a\left(1-e^{2}\right)-2r_{s}\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)},$ (26)

and

$\displaystyle\frac{J}{\left|A\right|^{\frac{1}{2}}}=\frac{a^{\frac{3}{2}}\left(1-e^{2}\right)}{\left(a\left(1-e^{2}\right)-2r_{s}\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)^{\frac{1}{2}}}$ (27)

Going back to equation (20), and because
$\frac{dr}{d\tau}=\left(\frac{dr}{d\phi}\right)\left(\frac{d\phi}{d\tau}\right)$,
replacing the expression of $\frac{d\phi}{d\tau}$ given by (15), we get:

$\displaystyle\left(\frac{dr}{d\phi}\right)^{2}\frac{1}{\gamma^{2}c^{2}r^{4}}\left(\frac{r_{s}\alpha}{r}E^{\prime}+\gamma
cJ\right)^{2}=A+\frac{r_{s}c^{2}}{r}-\frac{J^{2}}{r^{2}}+\frac{J^{2}r_{s}}{r^{3}}-\frac{2r_{s}\alpha}{cr^{3}}JE^{\prime}$
(28)

Leaving only first order terms in $\frac{\alpha}{r}$ we arrive to the
following equation:

$\displaystyle\left(\frac{dr}{d\phi}\right)^{2}\left(1+\frac{2r_{s}\alpha
E^{\prime}}{\gamma
rcJ}\right)=\frac{A}{J^{2}}r^{4}+\frac{r_{s}c^{2}}{J^{2}}r^{3}-r^{2}+\left(r_{s}-\frac{2r_{s}\alpha}{cJ}E^{\prime}\right)r$
(29)

that also can be written as:

$\displaystyle\left(\frac{dr}{d\phi}\right)^{2}\left(1+\frac{2r_{s}\alpha
E^{\prime}}{\gamma
rcJ}\right)=\frac{\left|A\right|}{J^{2}}\left(R_{a}-r\right)\left(r-R_{p}\right)\left(r-R^{\prime}_{o}\right)r.$
(30)

The advance of the perihelion, then will be given by the integral:

$\displaystyle\Delta\phi_{kerr}=\frac{2J}{\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\frac{\left(1+\frac{2r_{s}\alpha
E^{\prime}}{\gamma
rcJ}\right)^{1/2}dr}{\left[\left(R_{a}-r\right)\left(r-R_{p}\right)\left(r-R^{\prime}_{o}\right)r\right]^{1/2}}$
(31)

and keeping only terms of up to first order in $\frac{\alpha}{r}$:

$\displaystyle\Delta\phi_{kerr}=\frac{2J}{\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\frac{dr}{\left[\left(R_{a}-r\right)\left(r-R_{p}\right)\left(r-R^{\prime}_{o}\right)r\right]^{1/2}}\qquad\qquad\qquad\qquad\qquad\qquad\
$

$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{2r_{s}\alpha
E^{\prime}}{c\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\frac{dr}{r\gamma\left[\left(R_{a}-r\right)\left(r-R_{p}\right)\left(r-R^{\prime}_{o}\right)r\right]^{1/2}}$
(32)

that can be written as:

$\displaystyle\Delta\phi_{kerr}=\Delta\phi_{sch}(R^{\prime}_{o})+\Delta\phi_{2}(R^{\prime}_{o})$
(33)

where:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})=\frac{2J}{\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\frac{dr}{\left[\left(R_{a}-r\right)\left(r-R_{p}\right)\left(r-R^{\prime}_{o}\right)r\right]^{1/2}},$
(34)

and

$\displaystyle\Delta\phi_{2}(R^{\prime}_{o})=\frac{2r_{s}\alpha
E^{\prime}}{c\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\frac{dr}{\left(r-r_{s}\right)\left[\left(R_{a}-r\right)\left(r-R_{p}\right)\left(r-R^{\prime}_{o}\right)r\right]^{1/2}}$
(35)

The first term in equation (33) is the same that was obtained in a previous
paper MarinPoveda  using the Schwarzschild metric, but replacing $R_{o}$ by
$R^{\prime}_{o}$.

The parameter $\alpha$ is related to the spin of the black hole $s$, also
known as the kerr parameter through the relationship:

$\alpha=\frac{GM}{c^{2}}s,$

in such a way that the angular momentum of spin is:

$S_{z}=\frac{GM^{2}}{c}s,$ (36)

where $s$ is a dimensionless parameter that can take values between 0 and 1.
If $s$ was greater than 1, there would be no event horizons and the
singularity of $r=0$ would be naked, which is not allowed Ryder ; Wald .

##  III Calculation of $\Delta\phi_{sch}(R^{\prime}_{o})$

The first term in equation (33) also can be expressed as:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})=\frac{2J}{\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\frac{r^{-\frac{1}{2}}\left(1-\frac{R^{\prime}_{o}}{r}\right)^{-\frac{1}{2}}dr}{\left(\left(R_{a}-r\right)\left(r-R_{p}\right)r\right)^{\frac{1}{2}}},$
(37)

and because $R^{\prime}_{o}<<R_{p}<R_{a}$, we can expand around
$R^{\prime}_{o}$:

$\displaystyle\left(1-\frac{R^{\prime}_{0}}{r}\right)^{-\frac{1}{2}}=\sum_{n=1}^{\infty}\left(\begin{array}[]{c}-\frac{1}{2}\\\
n-1\end{array}\right)\left(-1\right)^{n-1}\frac{\left(R^{\prime}_{o}\right)^{n-1}}{r^{n-1}}.$
(40)

Then

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})=\frac{2J}{\left|A\right|^{1/2}}\sum_{n=1}^{\infty}\left(\begin{array}[]{c}-\frac{1}{2}\\\
n-1\end{array}\right)\left(-1\right)^{n-1}\left(R^{\prime}_{o}\right)^{n-1}I_{n},$
(43)

where:

$\displaystyle
I_{n}=\intop_{R_{p}}^{R_{a}}\frac{dr}{r^{n}\left(\left(R_{a}-r\right)\left(r-R_{p}\right)\right)^{\frac{1}{2}}}.$
(44)

The value of $I_{n}$ was calculated using complex integration in reference
MarinPoveda , where we obtained:

$\displaystyle
I_{n}=\frac{\pi(-1)^{n+1}}{a^{n}2^{n-1}\left(1-e^{2}\right)^{n-1/2}}\sum_{k=0}^{n-1}\left(\begin{array}[]{c}n-1\\\
k\end{array}\right)^{2}z_{1}^{n-1-k}z_{2}^{k}e^{n-1}$ (47)

where:

$\displaystyle z_{1}=-\frac{\left(1+\sqrt{1-e^{2}}\right)}{e},$ (48)

and

$\displaystyle z_{2}=-\frac{\left(1-\sqrt{1-e^{2}}\right)}{e},$ (49)

Defining the functions $Q_{n-1}\left(z_{1},z_{2}\right)$

$\displaystyle
Q_{n}\left(z_{1},z_{2}\right)=\sum_{k=0}^{n}\left(\begin{array}[]{c}n\\\
k\end{array}\right)^{2}z_{1}^{n-k}{z_{2}}^{k}e^{n},$ (52)

we can write:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})=\frac{2\pi
J}{\left|A\right|^{1/2}}\sum_{n=0}^{\infty}\left(\begin{array}[]{c}-\frac{1}{2}\\\
n\end{array}\right)\frac{\left(R^{\prime}_{o}\right)^{n}Q_{n}\left(z_{1},z_{2}\right)}{a^{n+1}2^{n}\left(1-e^{2}\right)^{\left(n+\frac{1}{2}\right)}}.$
(55)

In table (1) it is shown the first five functions $Q_{n}$.

Table 1: Values of the functions $Q_{n}$ Function | Expression
---|---
$Q_{0}$ | $1$
$Q_{1}$ | $-2$
$Q_{2}$ | $\left(4+2e^{2}\right)$
$Q_{3}$ | $-\left(8+12e^{2}\right)$
$Q_{4}$ | $\left(16+48e^{2}+6e^{4}\right)$
$Q_{5}$ | $-\left(32+160e^{2}+60e^{4}\right)$

Introducing the values of $R^{\prime}_{o}$ and
$\frac{J}{\left|A\right|^{1/2}}$ given by equations (26) and (27),
respectively, we get:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})=\frac{2\pi}{\left(1-2\epsilon\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)^{\frac{1}{2}}}\times\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\qquad\times\sum_{n=0}^{\infty}\left(\begin{array}[]{c}-\frac{1}{2}\\\
n\end{array}\right)\frac{\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)^{n}\epsilon^{n}}{2^{n}\left(1-2\epsilon\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)^{n}}Q_{n}\left(z_{1},z_{2}\right),$ (58)

where

$\displaystyle\epsilon\equiv\frac{r_{s}}{a\left(1-e^{2}\right)}=\frac{2GM}{a\left(1-e^{2}\right)c^{2}},$
(59)

and

$\displaystyle\left(\begin{array}[]{c}-\frac{1}{2}\\\
n\end{array}\right)=\frac{\left(-1\right)^{n}\left(2n\right)!}{2^{2n}\left(n!\right)^{2}}.$
(62)

##  IV Calculation of $\Delta\phi_{2}(R^{\prime}_{o})$

The integral $\Delta\phi_{2}(R^{\prime}_{o})$ also can be written as:

$\displaystyle\Delta\phi_{2}(R^{\prime}_{o})=\frac{2r_{s}\alpha
E^{\prime}}{c\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\frac{\left(1-\frac{2r_{s}+R^{\prime}_{o}}{r}+\frac{2R^{\prime}_{o}r_{s}+r_{s}^{2}}{r^{2}}-\frac{r_{s}^{2}R^{\prime}_{o}}{r^{3}}\right)^{-\frac{1}{2}}dr}{r^{2}\left[\left(R_{a}-r\right)\left(r-R_{p}\right)\right]^{1/2}}$
(63)

Let us call: $a_{1}=2r_{s}+R^{\prime}_{o}$,
$a_{2}=2R^{\prime}_{o}r_{s}+r_{s}^{2}$, and $a_{3}=r_{s}^{2}R^{\prime}_{o}$.
Performing the expansion:

$\displaystyle\left(1-\frac{a_{1}}{r}+\frac{a_{2}}{r^{2}}-\frac{a_{3}}{r^{3}}\right)^{-\frac{1}{2}}=\sum_{n=0}^{\infty}\left(\begin{array}[]{c}-\frac{1}{2}\\\
n\end{array}\right)\left(-\frac{a_{1}}{r}+\frac{a_{2}}{r^{2}}-\frac{a_{3}}{r^{3}}\right)^{n}=\sum_{n=0}^{\infty}\frac{c_{n}}{r^{n}},$
(66)

we can check that for example:

$\displaystyle c_{0}=1,$ (67) $\displaystyle c_{1}=\frac{a_{1}}{2},$ (68)
$\displaystyle c_{2}=\frac{1}{2}\left(\frac{3}{4}a_{1}^{2}-a_{2}\right),$ (69)
$\displaystyle
c_{3}=\frac{5}{16}a_{1}^{3}-\frac{3}{4}a_{1}a_{2}+\frac{1}{2}a_{3},$ (70)
$\displaystyle
c_{4}=\frac{35}{128}a_{1}^{4}+\frac{3}{4}a_{1}a_{3}-\frac{15}{16}a_{1}^{2}a_{2}+\frac{3}{8}a_{2}^{2},$
(71) $\displaystyle etc....$

In general the $c_{n}$ satisfy the following recurrence relationship:

$\displaystyle
c_{n+3}=\frac{1}{6+2n}\left(a_{1}c_{n+2}\left(5+2n\right)-2a_{2}c_{n+1}\left(2+n\right)+a_{3}c_{n}\left(3+2n\right)\right)$
(72)

In terms of the $c_{n}$ we can write (63) as:

$\displaystyle\qquad\qquad\Delta\phi_{2}(R^{\prime}_{o})=\frac{2r_{s}\alpha
E^{\prime}}{c\left|A\right|^{1/2}}\intop_{R_{p}}^{R_{a}}\sum_{n=0}^{\infty}\frac{c_{n}dr}{r^{n+2}\left[\left(R_{a}-r\right)\left(r-R_{p}\right)\right]^{1/2}}$
$\displaystyle=\frac{2r_{s}\alpha
E^{\prime}}{c\left|A\right|^{1/2}}\sum_{n=0}^{\infty}c_{n}I_{n+2}\qquad$ (73)

Replacing the value on $I_{n+2}$ given by (47) and the value of
$\frac{J}{\left|A\right|^{1/2}}$ given by (27) we get:

$\displaystyle\Delta\phi_{2}(R^{\prime}_{o})=\frac{\pi\epsilon}{2\left(1-2\epsilon\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)^{\frac{1}{2}}}\left(\frac{2\alpha
E^{\prime}}{cJ}\right)\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n+1}c_{n}Q_{n+1}\left(z_{1},z_{2}\right)\epsilon^{n}}{2^{n}r_{s}^{n}}$
(74)

##  V Expansion of $\Delta\phi_{sch}(R^{\prime}_{o})$ in terms of
$\epsilon^{*}\equiv\left(1-\frac{2\alpha E^{\prime}}{cJ}\right)\epsilon$

In terms of $\epsilon^{*}\equiv\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\epsilon$ (58) is:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})=\frac{2\pi}{\left(1-2\epsilon^{*}\right)^{\frac{1}{2}}}\
\times\sum_{n=0}^{\infty}\left(\begin{array}[]{c}-\frac{1}{2}\\\
n\end{array}\right)\frac{\left(\epsilon^{*}\right)^{n}}{2^{n}\left(1-2\epsilon^{*}\right)^{n}}Q_{n}\left(z_{1},z_{2}\right).$
(77)

We will expand $\Delta\phi_{sch}(R^{\prime}_{o})$ in terms of $\epsilon^{*}$.
Lets compute the first four terms of (77) to recover the expansion until third
order on $\epsilon^{*}$.

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})^{(3)}=\frac{2\pi}{\left(1-2\epsilon^{*}\right)^{1/2}}(Q_{0}\left(z_{1},z_{2}\right)-\left(\frac{1}{2}\right)\frac{Q_{1}\left(z_{1},z_{2}\right)}{2\left(1-2\epsilon^{*}\right)}\epsilon^{*}+\left(\frac{3}{8}\right)\frac{Q_{2}\left(z_{1},z_{2}\right)}{2^{2}\left(1-2\epsilon^{*}\right)^{2}}\left(\epsilon^{*}\right)^{2}$
$\displaystyle-\left(\frac{5}{16}\right)\frac{Q_{3}\left(z_{1},z_{2}\right)}{2^{3}\left(1-2\epsilon^{*}\right)^{3}}\left(\epsilon^{*}\right)^{3}+......).$
(78)

Replacing the values of the $Q_{i}$ ($i=0,1,2,3$) we have:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})^{(3)}=\frac{2\pi}{\left(1-2\epsilon^{*}\right)^{1/2}}(1+\frac{\epsilon^{*}}{2\left(1-2\epsilon^{*}\right)}+\left(\frac{3}{16}\right)\frac{\left(2+e^{2}\right)\left(\epsilon^{*}\right)^{2}}{\left(1-2\epsilon^{*}\right)^{2}}$
$\displaystyle+\left(\frac{5}{32}\right)\frac{\left(2+3e^{2}\right)\left(\epsilon^{*}\right)^{3}}{\left(1-2\epsilon^{*}\right)^{3}}+........)$
(79)

which in third orden in $\epsilon^{*}$ is reduced to:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})^{(3)}=2\pi\left(1+\frac{3}{2}\epsilon^{*}+\frac{\left(54+3e^{2}\right)}{16}\left(\epsilon^{*}\right)^{2}+\left(\frac{135}{16}+\frac{45}{32}e^{2}\right)\left(\epsilon^{*}\right)^{3}+....\right).$
(80)

In terms of $\epsilon$ we have:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})^{(3)}=2\pi(1+\frac{3}{2}\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\epsilon+\frac{3\left(18+e^{2}\right)}{16}\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)^{2}\epsilon^{2}$
$\displaystyle+\frac{45}{32}\left(6+e^{2}\right)\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)^{3}\epsilon^{3}+....).$ (81)

In general to order $m$ in $\epsilon^{*}$ we can write:

$\displaystyle\Delta\phi_{sch}(R^{\prime}_{o})^{(m)}=\frac{2\pi}{\left(1-2\epsilon^{*}\right)^{\frac{1}{2}}}\
\times\sum_{n=0}^{m}\left(\begin{array}[]{c}-\frac{1}{2}\\\
n\end{array}\right)\frac{\left(\epsilon^{*}\right)^{n}}{2^{n}\left(1-2\epsilon^{*}\right)^{n}}Q_{n}\left(z_{1},z_{2}\right).$
(84)

##  VI Expansion of $\Delta\phi_{2}(R^{\prime}_{o})$ in terms of $\epsilon$

To evaluate $\Delta\phi_{2}(R^{\prime}_{o})$ to third order in $\epsilon$ it
is sufficient to consider the first three terms in the sum given by equation
(74).

$\displaystyle\Delta\phi_{2}(R^{\prime}_{o})^{\left(3\right)}=\frac{\pi\epsilon}{2\left(1-2\epsilon\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)^{\frac{1}{2}}}\left(\frac{2\alpha
E^{\prime}}{cJ}\right)\left(-c_{0}Q_{1}+\frac{c_{1}Q_{2}\epsilon}{2r_{s}}-\frac{c_{2}Q_{3}\epsilon^{2}}{2^{2}r_{s}^{2}}\right),$
(85)

where

$c_{0}=1,$ (86) $c_{1}=\frac{2r_{s}+R^{\prime}_{o}}{2},$ (87)
$c_{2}=\frac{3}{8}\left(R^{\prime}_{o}\right)^{2}+\frac{1}{2}r_{s}R^{\prime}_{o}+r_{s}^{2}.$
(88)

If we introduce the values of the $Q_{i}$ in (85) we have:

$\displaystyle\Delta\phi_{2}(R^{\prime}_{o})^{\left(3\right)}=\frac{\pi\epsilon}{\left(1-2\epsilon\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)^{\frac{1}{2}}}\left(\frac{2\alpha
E^{\prime}}{cJ}\right)(1+\frac{\left(2r_{s}+R^{\prime}_{o}\right)\left(2+e^{2}\right)\epsilon}{4r_{s}}$
$\displaystyle+\frac{\left(3\left(R^{\prime}_{o}\right)^{2}+4r_{s}R^{\prime}_{o}+8r_{s}^{2}\right)\left(2+3e^{2}\right)\epsilon^{2}}{16r_{s}^{2}}),$
(89)

where

$\displaystyle\frac{\left(2r_{s}+R^{\prime}_{o}\right)}{r_{s}}=\frac{2+\left(1-4\epsilon\right)\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)}{\left(1-2\epsilon\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)},$ (90)

and

$\displaystyle\frac{\left(3\left(R^{\prime}_{o}\right)^{2}+4r_{s}R^{\prime}_{o}+8r_{s}^{2}\right)}{r_{s}^{2}}=\frac{8+4\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\left(1-8\epsilon\right)+\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)^{2}\left(3-8\epsilon+32\epsilon^{2}\right)}{\left(1-2\epsilon\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\right)^{2}}.$ (91)

Introducing (90) and (91) in (89), and employing the Taylor series (around
$x=0$): $\left(1-x\right)^{-\frac{1}{2}}=1+\frac{1}{2}x+\frac{3}{8}x^{2}+...$,
we obtain to third order in $\epsilon$:

$\displaystyle\Delta\phi_{2}(R^{\prime}_{o})^{\left(3\right)}=\pi\frac{2\alpha
E^{\prime}}{cJ}\epsilon+\frac{\pi}{4}\left(10+3e^{2}-\left(6+e^{2}\right)\frac{2\alpha
E^{\prime}}{cJ}\right)\frac{2\alpha E^{\prime}}{cJ}\epsilon^{2}$
$\displaystyle+\frac{\pi}{16}\left(86+53e^{2}-\left(116+54e^{2}\right)\frac{2\alpha
E^{\prime}}{cJ}+\left(38+13e^{2}\right)\left(\frac{2\alpha
E^{\prime}}{cJ}\right)^{2}\right)\frac{2\alpha E^{\prime}}{cJ}\epsilon^{3}$
(92)

##  VII Perihelion precession to third order in $\epsilon$

As the perihelion precession in a cycle is $\chi=\Delta\phi_{kerr}-2\pi$, we
have until third order in $\epsilon$:

$\displaystyle\chi^{\left(3\right)}=\Delta\phi_{sch}(R^{\prime}_{o})^{(3)}+\Delta\phi_{2}(R^{\prime}_{o})^{\left(3\right)}-2\pi,$
(93)

that give us:

$\displaystyle\chi^{\left(3\right)}=3\pi\left(1-\frac{4\alpha
E^{\prime}}{3cJ}\right)\epsilon+\left(\frac{\left(54+3e^{2}\right)}{8}-11\left(\frac{2\alpha
E^{\prime}}{cJ}\right)+\frac{\left(42+e^{2}\right)}{8}\left(\frac{2\alpha
E^{\prime}}{cJ}\right)^{2}\right)\pi\epsilon^{2}$
$\displaystyle+(\frac{45\left(6+e^{2}\right)}{16}-\frac{\left(362+41e^{2}\right)}{8}\left(\frac{2\alpha
E^{\prime}}{cJ}\right)+\frac{\left(694+81\right)}{16}\left(\frac{2\alpha
E^{\prime}}{cJ}\right)^{2}$
$\displaystyle-\frac{\left(29+4e^{2}\right)}{2}\left(\frac{2\alpha
E^{\prime}}{cJ}\right)^{3})\pi\epsilon^{3}.$ (94)

##  VIII Applications

To analyze how the spin of the central black hole changes the perihelion
precession, it was calculated until 3rd order (using equation (94)) for three
different binary systems. In table (2) are shown the results. The term
$\chi(\epsilon^{i})$ is the one that depends only on the i-th power of
$\epsilon$, since
$\chi^{(i)}=\chi(\epsilon^{1})+\chi(\epsilon^{2})+...+\chi(\epsilon^{i})$. The
expansion was done in terms of $\alpha$, that is a parameter with units of
length, but the experimental measurements are done in terms of the spin $s$.
The spin $s$ of the black hole is a dimensionless parameter that is related to
$\alpha$ by:

$\alpha=\frac{GM}{c^{2}}s=\frac{r_{s}}{2}s$ (95)

The spin of a central black hole in a binary system is very difficult to
measure. Nevertheless, some groups have measured it for the binary system
OJ287. This measurements can be seen in articles by Pihajoki and Valtonen
Pihajoki ; Valtonen . The problem is that the values obtained form those
measurements are different, however we worked with both values for the
numerical evaluations. Also we made the calculations for $s=0$.

For the calculations we need the values of the energy per unit mass and the
angular momentum per unit mass. For this, both values can be calculated
numerically using equations (21) and (22). These are the equations of motion
evaluated for the critical points $R_{a}$ (aphelion) and $R_{p}$ (perihelion).
As the equations are quadratic, there are two solutions for each parameter. We
took the negative value for the energy (that is necessary for the orbit to be
elliptical), and the positive value for the angular momentum, because we
considered a counterclockwise rotation.

Table 2: Perihelion advance in degrees per period for some binary systems, and for different values of the spin.  System | Sagittarius A*-S2 | OJ287 | H1821+643
---|---|---|---
$M(\times M_{\Theta})$ | $4.310\times 10^{6}$ | $1.830\times 10^{10}$ | $3.000\times 10^{10}$
$r_{s}(AU)$ | $0.085$ | $360.847$ | $591.553$
$a(AU)$ | $923.077$ | $11500$ | $40000$
$e$ | $0.870$ | $0.700$ | $0.900$
$\epsilon$ | $3.787\times 10^{-4}$ | $6.153\times 10^{-2}$ | $7.784\times 10^{-2}$
$s$ | $0$ | $0$ | $0$
$\chi(\epsilon)$ | $0.205^{\circ}$ | $33.223^{\circ}$ | $42.031^{\circ}$
$\chi(\epsilon^{2})$ | $(1.816\times 10^{-4})^{\circ}$ | $4.724^{\circ}$ | $7.692^{\circ}$
$\chi(\epsilon^{3})$ | $(1.858\times 10^{-7})^{\circ}$ | $0.765^{\circ}$ | $1.625^{\circ}$
$\chi^{(3)}$ | $0.205^{\circ}$ | $38.713^{\circ}$ | $51.349^{\circ}$
$s$ | $0.280$ | $0.280$ | $0.280$
$\chi(\epsilon)$ | $0.206^{\circ}$ | $35.249^{\circ}$ | $46.230^{\circ}$
$\chi(\epsilon^{2})$ | $(1.837\times 10^{-4})^{\circ}$ | $5.440^{\circ}$ | $9.622^{\circ}$
$\chi(\epsilon^{3})$ | $(1.895\times 10^{-7})^{\circ}$ | $0.965^{\circ}$ | $2.349^{\circ}$
$\chi^{(3)}$ | $0.206^{\circ}$ | $41.654^{\circ}$ | $58.203^{\circ}$
$s$ | $0.313$ | $0.313$ | $0.313$
$\chi(\epsilon)$ | $0.206^{\circ}$ | $35.488^{\circ}$ | $46.727^{\circ}$
$\chi(\epsilon^{2})$ | $(1.841\times 10^{-4})^{\circ}$ | $5.528^{\circ}$ | $9.866^{\circ}$
$\chi(\epsilon^{3})$ | $(1.899\times 10^{-7})^{\circ}$ | $0.991^{\circ}$ | $2.448^{\circ}$
$\chi^{(3)}$ | $0.206^{\circ}$ | $42.007^{\circ}$ | $59.420^{\circ}$

In table (2) it can be seen the values for $s=0$ for the systems Sagittarius
A*-S2, OJ287, H1821+643. These values agree with those calculated in a
previous paper MarinPoveda  using the Schwarzschild metric. These results are
in concordance with the fact that the Kerr space-time transforms in
Schwarzschild space-time when $s=0$.

Then, for other values of $s$, the perihelion precession increases with $s$.
Also, it is important to recall that we considered that $s$ is positive, i.e,
the central black hole rotates in the same direction of the movement of the
secondary massive object. We have used the values of the spin of OJ287 with
all the systems to see how the perihelion precession changes with it because
we do not have information about the values for those systems.

For OJ287, the experimental accepted value for the perihelion precession is
$39.1^{\circ}$ per period. The theoretical calculation (for $s=0.313$)
predicts a higher value of $42.0^{\circ}$. Without considering the spin, the
theoretical value is $38.7^{\circ}$. It was expected that the introduction of
the spin would reduce the error with respect to the experimental value, but
instead it exceeded this value. Nevertheless, there is another correction that
is expected to reduce the value of the perihelion precession. This correction
consist in taking into account the lost of energy in the gravitational
radiation. That correction will be done in further works.

##  IX Conclusions

In this paper we have considered both Schwarzschild and Kerr metrics to take
into account the spin effect in the calculation of the perihelion precession
in binary systems like OJ287, Sagittarius A*-S2, and H1821+643. We have
developed the expansion in terms of both $\epsilon\equiv
r_{s}/\left(a(1-e^{2})\right)$ and $\epsilon^{*}\equiv\left(1-\frac{2\alpha
E^{\prime}}{cJ}\right)\epsilon$, where $E^{\prime}$ and $J$ are the energy and
angular momentum per unit mass and $\alpha$ is a factor proportional to the
black hole spin. Here we have used the notation
$\chi\left(\epsilon^{n}\right)$ for the contribution of the $n^{th}$ term and
$\chi^{\left(n\right)}$ for the complete expansion until $n^{th}$ term.

Now, to see if the spin correction is relevant for massive objects, we
calculated the perihelion advance for three binary systems, Sagittarius A*-S2,
OJ287 and H1821+643. In table (2) are shown these calculations performed using
equation (94) for different orders.

Sagitarius A* is a bright and very compact radio source located at the center
of the Milky Way. There is great evidence that Sagittarius A* is a
supermassive black hole wtith a mass approximately equal to $4\times
10^{6}M_{\Theta}$ . We took star S2, because is the one that presents a very
peculiar orbit. As it can be seen in table (2), if we compare the value of
$\chi^{(3)}$ for $s=0$ that is $0.205^{\circ}$ with the corresponding values
for $s=0.280$ and $s=0.313$ that is $0.206^{\circ}$, we can see that the
difference is not significant. It is worth mentioning that the values of
$\chi\left(\epsilon^{n}\right)$ and $\chi^{\left(n\right)}$ corresponding to
$s=0$ are agree with those calculated in a earlier paper MarinPoveda  using
the Schwarzschild metric.

OJ 287 is a binary system of black holes located 3.500 million light years
from Earth having a total mass of around $1.845\times 10^{10}M_{\Theta}$. It
can be seen that between the first and the second order terms
($\chi\left(\epsilon\right)$ and $\chi\left(\epsilon^{2}\right)$) there is a
difference of approximately $5.5^{\circ}$. For higher orders, the difference
is less than $1^{\circ}$. At the third order the contribution to the
perihelion precession in a cycle is $0.965^{\circ}$ in our expansion for
$s=0.280$ compared with the value of $0.765^{\circ}$ obtained in a earlier
paper MarinPoveda  taking $s=0$. Then, the total perihelion precession is
$41.654^{\circ}$. For $s=0.313$ the perihelion precession is slightly
different (approximately $42^{\circ}$). Something interesting about our
expansion is that it begins to stabilize taking into account higher order
corrections around $42^{\circ}$. Then, equation (94)) gives an important
correction to the perihelion precession. It is important to mention that the
gravitational radiation of the system (that affects directly to the values of
the period, the radial distance and the eccentricity of the orbit Shreya ),
could introduce important modifications in the perihelion advance equation.

For H1821+643 system , that is a very massive black hole with a mass of
$3\times 10^{10}M_{\Theta}$, the orbital parameters of the gravitational
companion are not known, so we used random parameters. In this system, the
corrections are very important. For example, taking $s=0.313$, the second
order contribution $\chi\left(\epsilon^{2}\right)$ is $9.866^{\circ}$, while
the third order contribution $\chi\left(\epsilon^{3}\right)$ is
$2.448^{\circ}$ .

Taking into account all these results, we can conclude that this is an
excellent approach to calculate the perihelion precession for binary systems
as it can be calculated until any order in the third root of the motion
equation $R^{\prime}_{o}$ (see equation (26)). With other methods as
perturbation theory, the calculations are more complicated and due to the
approximations that are made it is easy to make mistakes.

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