The Lense-Thirring Effect in the Anomalistic Period of Celestial Bodies

In the weak field and slow motion approximation, the general relativistic field equations are linearized, resembling those of the electromagnetic theory. In a way analogous to that of a moving charge generating a magnetic field, a mass-energy current can produce a gravitomagnetic field. In this contribution, the motion of a secondary celestial body is studied under the influence of the gravitomagnetic force generated by a spherical primary. More specifically, two equations are derived to approximate the periastron time rate of change and its total variation over one revolution (i.e., the difference between the anomalistic period and the Keplerian period). Kinematically, this influence results to an apsidal motion. The aforementioned quantities are numerically estimated for Mercury, the companion star of the pulsar PSR 1913+16, the companion planet of the star HD 80606 and the artificial Earth satellite GRACE-A. The case of the artificial Earth satellite GRACE-A is also considered, but the results present a low degree of reliability from a practical standpoint.


INTRODUCTION
In the weak-field linearization of the general relativistic field equations, the theory predicts that the gravitational field of a rotating primary body results to a magnetic-type force that is called gravitomagnetic. This force affects secondary bodies, gyroscopes and clocks that move around this primary as well as electromagnetic waves (Schafer, 2009). Lense and Thirring (1918) worked on the gravitomagnetic influence on the motion of test particles orbiting a slow rotating primary mass. They particularly interested on celestial systems. They found that this influence, hereinafter called Lense-thirring effect, results to precession in the argument of the perigee and the ascending node of the orbits of the secondaries.
Far from the rotating body, the gravitomagnetic field can be expressed as (Mashhoon et al., 2001): where, G is the constant of universal gravitation, c is the speed of light, r is the unit vector along the position vector r r and S v is the angular momentum vector of the rotating body. The field indicated by Equation (1) equals to that of a magnetic dipole whose moment is Equation 2 (Iorio et al., 2011): Then, a secondary, that is moving with a velocity v r in this field, is affected by a non-central acceleration that is given by: In stronger gravitational fields, like those of neutron stars and black holes, higher order corrections in v / c r Science Publications AJSS become important for the orbits of test particles, too. These kinds of scenarios have been studied by Capozziello et al. (2009) andalso, Schafer (2009).
The time rate of change, which is caused by the Lense-Thirring effect, in the argument of the periastron and the ascending node of the secondary are given by the following relations Equation 4 and 5: where, a, e, i represent the semimajor axis, the eccentricity and the inclination of the orbit. The radial, transverse and normal to the orbital plane components of the acceleration given in Equation 3 are Equation 6-8 (Iorio et al., 2011): where, f is the true anomaly and ξ is equal to Equation 9: Here M denotes the mass of the primary. In this study, we deal with the Lense-Thirring effect on the relative orbit of the secondary body with respect to the primary one. In other words, we reduce the initial two-body problem to a central-force problem. Within this framework, We establish the periastron time rate of change induced by this effect. Then, we determine the perturbation in the anomalistic period (the time interval elapsed between two successive transits of the secondary at the periastron) via the difference between this one and that predicted by the Kepler's theory (Mioc and Radu, 1979;Haranas et al., 2011). Numerical applications of our formulae are given for the motions of Mercury, the companion star of the pulsar PSR 1913+16, the companion planet of the star HD 80606 as well as the artificial Earth satellite GRACE-A.

RATE OF CHANGE AND VARIATION PER REVOLUTION OF THE PERIASTRON TIME
Consider the unperturbed relative orbit of the secondary, obviously a Keplerian ellipse. Let n be its mean motion and M the mean anomaly. First, we will express the rate of change of the periastron time T 0 in terms of the true anomaly f. The well-known relation Equation 10: Connects M to T 0 . Here t denotes the time variable. We differentiate Equation (3) with respect to t to obtain: The rate of change of the mean motion dn/dt can be found by using Kepler's third law. On the unperturbed Keplerian ellipse this law is expressed as follows Equation 12 and In the presence of a perturbation, the rates of change of the orbital elements can be found by using Gauss' planetary equations. In our case and for the semimajor axis, the mean anomaly, the argument of the periastron and the ascending node, they read: 2r cosf sin f A dt n a n ae na 1 e 1 e r 1 sin f n ae a 1 e Next, we express Equation (19) in terms of the eccentric anomaly E by using the well-known relations Equation 20-24 (Murray and Dermott, 1999):  (20) and (24) To proceed with the integration we expand in power series of the eccentricity the following terms that appear in Equation (27) where, ℜ and w denote the radius and the angular velocity of the primary, respectively. Ii is seen that ∆T 0 is proportional to M 3/2 ℜ 2 w. This is how the anomalistic period change of the secondary depends on the characteristics of the primary.

NUMERICAL RESULTS
First, we proceed with the calculation of the change of the perihelion time of Mercury. We have used for the orbital parameters of this planet the following values: a = 57909083km, e = 0.205, I = 7.004°, n = 8.26×10 -7 rad/s. For Sun we have used that. M = 1.99×10 30 kg, w = 2.863×10 −6 rad/s, while, supposing that this star is a spherical body, its radius is, approximately, ℜ = 6.96×10 8 km. Τhen, Equation (32) Figure 2 depicts the corresponding variation if its inclination is considered to be equal to 0.
Next, we consider the case of the pulsar PSR 1913+16 and its companion star. For the pulsar we have used that m = 2.870×10 30 kg, w = 0.454881162rad/s and ℜ = 9.74×10 3 km, while for the orbital parameters the companion a = 1.950×10 6 km, e = 0.617, i = 45°and n = 1.60608×10 -4 rad/s. Then, the change of the anomalistic period of this companion is found to be Equation 40: Another test case will be the well known extra-solar planet b, a companion planet of the star HD 80606 in the constellation of Ursa Major. This planet is a superjovian planet whose orbital parameters (http://exoplanet.hannorein.de/system.php?id = HD+80606+b) are approximately: a = 6.795×10 7 km, e = 0.933, i = 89.285° and n = 4.0×10 −8 rad/s. Also, the mass, the angular velocity and the radius of HD 80606 are M= 1.791×10 30 kg, w = 1.168×10 −3 rad/s and ℜ = 5.99×10 −8 km. Then, the change of the anomalistic period of the planet b will be Equation 43: Comparing the values of ∆T 0 for the polar and equilateral orbit in any of the two last example cases, it seems that this change for polar orbits is, in absolute value, bigger than that for equilateral ones.
In the case of GRACE-A, an artificial satellite of Earth, we have that its orbital elements are a = 6876.4816 km, e = 0.00040989, i = 89.025446° and n = 0.001100118 rad/s (http://www.csr.utexas.edu/grace/). For Earth we have that M = 5.9736×10 24 kg and w = 7.292×10 −5 rad/s and ℜ = 638.1363 km. Then, for GRACE-A, the change of the anomalistic period is Equation 46: But we must say that, practically, these results have a low degree of reliability, because of the very small eccentricity of this satellite. It is known that for quasicircular orbits the position of the periastron (hence the periastron passage time) cannot be accurately determined. However our calculations are of some interest regarding the magnitude order of the perigee passage time variation.

SUMMARY AND CONCLUDING REMARKS
We use the components of the gravitomagnetic force produced by a primary celestial body that gives rise to the Lense-Thirring effect in order to derive an eccentric anomalydependent equation that estimates the rate of change of the periastron time T 0 of a secondary body orbiting this primary. By using the integral of this equation over a whole revolution, we have found that the Lense-Thirring effect contributes to an advance or a recess of the periastron time, depending on the inclination and eccentricity of the secondary. A variation for T 0 of the order of microseconds may be detectable by today's technology. This variation was estimated for some concrete astronomical cases. Mercury exhibits an advance of 98.24 µs, the PSR 1913+16 companion star an advance of 0.061 µs, the planet HD 80606 b an advance of 3.91 µs and finally in the case of GRACE-A this advancing effect is equal to 0.1867 µs. The presented results can constitute a possible test for the action of the Lense-Thirring force on the solar system bodies, or other celestial objects. Of course, this is not the only effect to be considered. For example, other Science Publications AJSS effects like general relativistic and quantum effects can be also considered. This is another topic for us to deal with in the nearest future.