108 The Open Astronomy Journal, 2011, 4, (Suppl 1-M7) 108-150 Open Access The Newtonian and Relativistic Theory of Orbits and the Emission of Gravitational Waves

Mariafelicia De Laurentis1,2,*

1Dipartimento di Scienze Fisiche, Universitá di Napoli " Federico II",Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy 2INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy

Abstract: This review paper is devoted to the theory of orbits. We start with the discussion of the Newtonian problem of motion then we consider the relativistic problem of motion, in particular the post-Newtonian (PN) approximation and the further gravitomagnetic corrections. Finally by a classification of orbits in accordance with the conditions of motion, we calculate the gravitational waves luminosity for different types of stellar encounters and orbits. Keywords: Theroy of orbits, gravitational waves, post-newtonian (PN) approximation, LISA, transverse traceless, Elliptical orbits.

1. INTRODUCTION of General Relativity (GR). Indeed, the PN approximation method, developed by Einstein himself [1], Droste and de Dynamics of standard and compact astrophysical bodies could be aimed to unravelling the information contained in Sitter [2, 3] within one year after the publication of GR, led the gravitational wave (GW) signals. Several situations are to the predictions of i) the relativistic advance of perihelion of planets, ii) the gravitational redshift, iii) the deflection possible such as GWs emitted by coalescing compact binaries–systems of neutron stars (NS), black holes (BH) of light, iv) the relativistic precession of the Moon orbit, that driven into coalescence by emission of gravitational are the so–called "classical" tests of GR. radiation (considered in the inspiralling, merging and ring- On the other hand, as emphasized by Eisenstaedt [4], the down phases, respectively), hard stellar encounters, and use of PN approximation method has had, from a conceptual other high-energy phenomena where GW emission is point of view, the adverse side-effect of introducing expected. Furthermore the signature of GWs can be always implicitly a 'neo-Newtonian' interpretation of GR. Indeed, determined by the relative motion of the sources. In this technically this approximation method follows the review paper, we want to discuss the problem of how the Newtonian way of tackling gravitational problems as closely waveform and the emission of GWs depend on the relative as possible. But this technical reduction of Einstein's theory motions in Newtonian, Relativistic and post-Relativistic regimes as for example situations where gravitomagnetic into the Procrustean bed of Newton's theory surreptitiously corrections have to be considered in the orbital motion. entails a corresponding conceptual reduction: the Einstenian problem of motion is conceived within the Newtonian As a first consideration, we have to say that the problem framework of an "absolute" coordinate space and an of motion, i.e. the problem of describing the dynamics of "absolute" coordinate time. However, some recent gravitationally interacting bodies, is the cardinal problem of developments oblige us to reconsider the problem of motion any theory of gravity. From the publication of Newton's within Einstein's theory. On the other hand, the discovery of Principia to the beginning of the twentieth century, this the binary pulsar PSR 1913+16 by Hulse and Taylor in 1974 problem has been thoroughly investigated within the [5], and its continuous observation by Taylor and coworkers framework of Newton's dynamics. This approach led to the (see references [6, 7]), led to an impressively accurate formulation of many concepts and theoretical tools which have been applied to other fields of physics. As a tracking of the orbital motion of a NS in a binary system. consequence, the relationship between Einstein's and This means that it is worth reconsidering in detail, i.e. at its Newton's theories of gravity has been, and still is, very foundation, the problem of motion also in relation to the peculiar. On the one hand, from a theoretical point of view, problem of generation and detection of GWs. In other words, the existence of Newton's theory facilitated the early the motion of sources could give further signatures to GWs development of Einstein's theory by suggesting an and then it has to be carefully reconsidered [8]. approximation method (called post-Newtonian (PN)) which The first part of this review paper is devoted to the theory allowed to draw very soon some observational consequences of orbits. The most natural way to undertake this task is starting with the discussion of the Newtonian problem of motion then we consider the relativistic problem of motion, *Address correspondence to this author at the INFN Sez. di Napoli, Compl. in particular the PN approximation and the further Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy; Tel: 0039081676881; E-mail:[email protected] gravitomagnetic corrections.

1874-3811/11 2011 Bentham Open The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 109

The theory of orbits can be connected to GWs since accurate estimate of the mass and the size of the central studies of binary systems prove, beyond reasonable doubts, object: we have (2.61± 0.76)
106 M concentrated within a that such a form of radiation has to exist. Detecting the
radius of (about 30 light-days) [30, 31]. More waves directly and exploiting them could result a very 0.016 pc impressive way to study astrophysical objects. In other precisely, in [30], it is described a campaign of observations words, the detection of GWs could give rise to the so-called where velocity measurements in the central arcsec2 are Gravitational Astronomy. extremely accurate. Then from this bulk of data, considering In view of this achievement, it is relevant to stress that a field of resolved stars whose proper motions are accurately GW science has entered a new era. Experimentally 1, several known, one can classify orbital motions and deduce, in ground-based laser-interferometer GW detectors ( kHz) 10
1 principle, the rate of production of GWs according to the have been built in the United States (LIGO) [9], Europe different types of orbits. (VIRGO and GEO) [10, 11] and Japan (TAMA) [12], and are now taking data at designed sensitivity. These issues motivate this review paper in which, by a classification of orbits in accordance with the conditions of A laser interferometer space antenna (LISA) [13] motion, we want to calculate the GW luminosity for different
4
2 ( 10
10 Hz) might fly within the next decade. types of stellar encounters and orbits (see also [32, 33]). From a theoretical point of view, last years have Following the method outlined in [34, 35], we investigate witnessed numerous major advances. Concerning the most the GW emission by binary systems in the quadrupole promising GW sources for ground-based and space-based approximation considering bounded (circular or elliptical) detectors, i.e. binary systems composed of NS, BHs, our and unbounded (parabolic or hyperbolic) orbits. Obviously, understanding of the relativistic two-body problem, and the the main parameter is the approaching energy of the stars in consequent GW-generation problem, has improved the system (see also [36] and references therein). We expect significantly. that gravitational waves are emitted with a "peculiar" Knowledge has also progressed on the problem of motion signature related to the encounter-type: such a signature has of a point particle in curved spacetime when the emission of to be a "burst" wave-form with a maximum in GWs is taken into account (non-geodesic motion) [14, 15]. correspondence of the periastron distance. The problem of Solving this problem is of considerable importance for bremsstrahlung-like gravitational wave emission has been predicting very accurate waveforms emitted by extreme studied in detail by Kovacs and Thorne [37] by considering mass-ratio binaries, which are among the most promising stars interacting on unbounded and bounded orbits. In this sources for LISA [16]. review paper, we face this problem discussing in detail the dynamics of such a phenomenon which could greatly The GW community working at the interface between the improve the statistics of possible GW sources. For further theory and the experiment has provided templates [17-19] details see also [26, 38-50]. for binaries and developed robust algorithms [20, 21] for pulsars and other GW-sources observable with ground-based The review is organized as follows. In Part I, as we said, and space-based interferometers. The joined work of data we discuss the theory of orbits. In Sec.2, we start with the analysts and experimentalists has established astrophysically Newtonian theory of orbits and discuss the main features of significant upper limits for several GW sources [22-24] and stellar encounters by classifying trajectories. Sec.3 is is now eagerly waiting for the first detection. devoted to orbits with relativistic corrections. A method for solving the equations of motion of binary systems at the first In this research framework, searching for criteria to PN-approximation is reviewed. The aim is to express the classify how sources collide and interact is of fundamental solution in a quasi-Newtonian form. In the Sec.4, we study importance. A first rough criterion can be the classification higher order relativistic corrections to the orbital motion of stellar encounters in collisional, as in the globular considering gravitomagnetic effects. We discuss in details clusters, and in collisionless as in the galaxies [25]. A how such corrections come out by taking into account fundamental parameter is the richness and the density of the "magnetic" components in the weak field limit of stellar system and then, obviously, we expect a large gravitational field. Finally, the orbital structure and the production of GWs in rich and dense systems. stability conditions are discussed giving numerical examples. Systems with these features are the globular clusters and Beside the standard periastron corrections of GR, a new 3 the galaxy centers. In particular, one can take into account nutation effect have to be considered thanks to c
the stars (early-type and late-type) which are around our corrections. The transition to a chaotic behavior strictly Galactic Center, e.g. Sagittarius A* ( SgrA* ) which could be depends on the initial conditions. The orbital phase space very interesting targets for the above mentioned ground- portrait is discussed. based and space-based detectors. Part II is devoted to the production and signature of In recent years, detailed information has been achieved for gravitational waves. We start, in Sec. 5, by deriving the wave kinematics and dynamics of stars moving in the gravitational equation in linearized gravity and discuss the gauge field of such a central object. The statistical properties of properties of GWs. Sec.6 is devoted to the problems of generation, emission and interaction of GWs with a detector. spatial and kinematical distributions are of particular interest In Sect. 7, we discuss the problem of GW luminosity and (see e.g. [27-29]). Using them, it is possible to give a quite emission from binary systems moving along Newtonian orbits. The quadrupole approximation is assumed. Sec. 8 is 1GW experiments started with the pioneering work of Joseph Weber at devoted to the same problem taking into account relativistic Maryland in the 60s 110 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis motion. In Sec. 9, also gravitomagnetic effects on the orbits hence p  p = p
m v and hence, with M = m + m , and the emission are considered. In Sec. 10, as an i i i i 1 2 outstanding application of the above considerations, we P  P = P
Mv, derive the expected rate of events from the Galactic Center. Due to the peculiar structure of this stellar system, it can be P considered a privileged target from where GWs could be and if we choose v = , then the total momentum is equal M detected and classified. Discussion, concluding remarks and to zero in the primed frame. We also note that the perspectives are given in Sec. 11. gravitational force is invariant under the Galilei transformation, since it depends only on the difference PART I r
r . Thus let us from now on work in the primed frame, THEORY OF ORBITS 1 2 but drop the primes for convenience of notation. We can II. Newtonian Orbits now replace the original equations of motion with the equivalent ones, We want to describe, as accurately as possible, the dynamics of a system of two bodies, gravitationally dp m m P =0, = G 1 2 r, interacting, each one having finite dimensions. Each body
3 exerts a conservative, central force on the other and no other dt r external forces are considered assuming the system as where r = r
r , r =| r |, and p = p
p . Next we isolated from the rest of the universe. Then, we first take into 1 2 1 2 account the non-relativistic theory of orbits since stellar introduce the position vector R of the center of mass of the systems, also if at high densities and constituted by compact system: objects, can be usually assumed in Newtonian regime. In m r + m r most cases, the real situation is more complicated. R = 1 1 2 2 , Nevertheless, in all cases, it is an excellent starting m1 + m2 approximation to treat the two bodies of interest as being isolated from outside interactions. We give here a self- hence contained summary of the well-known orbital types [25, 26] dR which will be extremely useful for the further discussion. P = M , dt A. Equations of Motion and Conservation Laws and hence from P =0 we have Newton's equations of motion for two particles of masses m and m , located at r and r , respectively, and R = const. 1 2 1 2 interacting by gravitational attraction are, in the absence of external forces, and we can carry out a translation of the origin of our coordinate frame such that R =0. The coordinate frame we dp m m have arrived at is called center-of-mass frame (CMS). We 1 = G 1 2 (r r ),
3 1
2 can also see now that dt | r
r | 1 2 dr1 dr dp m m p = p1 = m1 = μ , 2 = +G 1 2 (r
r ), (2.1) dt dt dt | r
r |3 1 2 1 2 m m where μ = 1 2 is the reduced mass of the system, and dri m m where p = m is the momentum of particle i , (i =1,2), 1 + 2 i i dt hence the equation of motion can be cast in the form and G is Newtonian gravitational constant. d 2r μ M d μ =
G rˆ, (2.2) ( p p )=0, 2 2 1 + 2 dt r dt r or, with Ppp= 12+ denoting the total momentum of the where we have defined the radial unit vector rˆ = . We can two body system, r get two more conservation laws if we take the scalar product P = const. dr Thus we have found a first conservation law, namely the of Eq. (2) with and its vector product with r . The scalar dt conservation of the total momentum of a two-body system in dr the absence of external forces. We can make use of this by product with gives on the left-hand side carrying out a Galilei transformation to another inertial dt frame in which the total momentum is equal to zero. Indeed, 2 let us apply the transformation dr d 2r 1 d  dr 
=   , dt dt 2 2 dt dt r  r= r
vt, i =1,2   i i i and on the right-hand side we have The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 111

rˆ dr d 1 p2 L2   =
 , r +
= E, r 2 dt dt r 2 2 r   μ 2μr hence μ or, with pr = dt , 2 dr d  p   =0, 
 2 dt 2μ r  1  dr  L2  μ +
= E. (2.5) 2  dt  2 r 2 r where
= GμM. This implies that the expression in brackets   μ is conserved, i.e. Looking back at our starting point, Eq. (2.1), we reduce p2  the dimensionality of our problem: from the simultaneous
= E = const. (2.3) differential equations of six functions of time, namely the six 2μ r components of the position vectors r1 and r2 , we reduce to a Here the first term is the kinetic energy, the second term is pair of simultaneous differential equations for the polar the potential energy, and the sum of kinetic energy and coordinates and (t) these equations contain two r(t)
E potential energy is the total energy , which is a constant of constants of motion, the total energy E and angular motion. Now take the cross product of Eq. (2) with r : on the momentum L . Then a mass m1 is moving in the right-hand side, we get the cross product of collinear vectors, which is equal to zero, hence gravitational potential
generated by a second mass m2 . The vector radius and the polar angle depend on the time as a d  dr  d consequence of the star motion, i.e. r = r(t) and = (t) . r
μ r
 = (r
p)=0,

dt  dt  dt With this choice, the velocity v of the mass m1 can be and hence, if we define the angular momentum L by parameterized as L = r
p, v = v rˆ + v
ˆ , r
we get the result where the radial and tangent components of the velocity are, respectively, dL =0, dr d
dt v = , v = r . r
or dt dt L = const, We can split the kinetic energy into two terms where, due to the conservation of angular momentum, the second one is i.e. conservation of angular momentum. An immediate a function of r only. An effective potential energy Veff , consequence of this conservation law is that the radius vector L2  r always stays in one plane, namely the plane perpendicular V =
, eff 2 r 2 r to L . This implies that we can without loss of generality μ choose this plane as the ()xy coordinate plane. The vector r is immediately defined. The first term corresponds to a is then a two-dimensional vector, repulsive force, called the angular momentum barrier. The second term is the gravitational attraction. The interplay r =(x, y)=rrˆ. rˆ =(cos
,sin
), between attraction and repulsion is such that the effective where we have defined the polar angle
. With this notation potential energy has a minimum. Indeed, differentiating with we can express the magnitude of angular momentum as L2 respect to r one finds that the minimum lies at r = and 0
μ 2 d
L = r , (2.4) that dt 2 The conservation of angular momentum can be used to min μ V =
. simplify the equation of motion (2.3). To do this we note that eff 2 2L L2 =(r p)2 = r 2 p2 (r p)2 , Therefore, since the radial part of kinetic energy, 

2 hence 1 dr Kr = μ   , 2  dt  L2 p2 = + p2 , r 2 r is non-negative, the total energy must be not less than V min , eff where prpr = ˆ
is the radial component of momentum. i.e. Substituting into Eq. (2.3) then gives 112 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

Table1. Orbits in Newtonian Regime Classified by the Approaching which is the canonical form of conic sections in polar Energy coordinates [51]. The constant C and
are two integration constants of the second order differential equation (2.8). The C =0 Circular Orbits solution (2.9) must satisfy the first order differential equation E = Emin (2.7). Substituting (2.9) into (2.7) we find, after a few
μ E < E <0 elliptic orbits algebra, 0< C < min 2 L 2 2 2μE 
μ 
μ E =0 parabolic orbits C = +   , (2.10) C = 2 2 2 L  L  L and therefore, taking account of Eq. (IIA), we get C 2
0 .
μ E >0, hyperbolic orbits C > 2 This implies the four kinds of orbits given in Table IIA and L in Fig. (1).

μ 2 E E = .
min  2 2L The equal sign corresponds to the radial motion. For E < E <0, the trajectory lies between a smallest value min r and greatest value r which can be found from the min max condition E = V , i.e. eff

2     L2 r =
±   + , {min,max} 2E 2E 2μE   where the upper (lower) sign corresponds to r (r ). max min Fig. (1). Newtonian paths: in black line we have hyperbolic path, in Only for E >0, the upper sign gives an acceptable value; blue line we have parabolic path, in red line the elliptical path and the second root is negative and must be rejected. Let us now in ciano the circular path. proceed in solving the differential equations (2.4) and (2.5). We have B. Circular Orbits u =0 dr dr d L dr L d  1 Circular motion corresponds to the condition
= = = , (2.6) 2 2
from which one find r = L / where V has its dt d dt d d  r  0 μ
eff  μr  μ    minimum. We also note that the expression for r0 together and defining, as standard, the auxiliary variable u =1/r , Eq. (2.5) takes the form with Eq. (II A) gives  2 2 2μ 2μE r = . (2.11) u + u
u = , (2.7) 0
2 2 2E L L min where u = du / d and we have divided by L2 /2 . Thus the two bodies move in concentric circles with 
μ Differentiating with respect to
, we get radii, inversely proportional to their masses and are always in opposition.  μ  uu + u
=0, L2 C. Elliptical Orbits   For 0< C < μ
/ L2 , r remains finite for all values of hence either u
=0, corresponding to the circular motion, or . Since r( 2 )=r( ) , the trajectory is closed and it is
μ
 +
 u u = , (2.8)  + 2 an ellipse. If one chooses =0, the major axis of the ellipse L
corresponds to . We get
=0 which has the solution
1 μ  μ r = r = + C , u = + C cos  +
, =0 min  2 2 ()  L  L or, reverting the variable, and

1 1   μ  μ r = r = C , r = + C cos  + , (2.9) = max  L2 () 
L2     The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 113

and the trajectory is a hyperbola. Such trajectories and since rmax + rmin =2a , where a is the semi-major axis of correspond to non-returning objects. Let us consider a semi- the ellipse, one obtains axis a and F as variable, analogous to the elliptic eccentric
1  2  anomaly E . The hyperbolic orbit is defined also by μ μ 2 a = r = r = C  , =0 min 2 2  + r = a( cosh F
1), L L    hence, there is the following relation between F and the C can be eliminated from the latter equation and Eq. (2.10) angle
: and then l
a( cosh F
1)  cos= . a =
, (2.12) a( cosh F
1) 2E Finally the parabolic orbit can be defined by the another Furthermore, if we denote the distance r by l , the 
=/2 relation (see [52]) so-called semi-latus rectum or the parameter of the ellipse, P2 we get r= , 2 L2 l = , (2.13) where P is a parameter. In this case
μ 2l
P2 cos = . and hence the equation of the trajectory  2 P l As we will discuss below, this classification of orbital r = , (2.14) 1+ cos
motions can reveal extremely useful for the waveform signature of gravitational radiation. Let us now take into account the relativistic theory of orbits. 1
l where  = is the eccentricity of the ellipse. If we a III. RELATIVISTIC ORBITS consider the major semiaxis of orbit a Eq. (2.12) and the eccentric anomaly , the orbit can be written also as (see As we have seen in the above Section, the non-relativistic
two bodies problem consists in two sub-problems: [52]) 1. Deriving the equations of orbital motion for two r = a(1
 cos E), gravitationally interacting extended bodies, this equation, known as Kepler's equation, is transcendental 2. Solving these equations of motion. in
, and the solution for this quantity cannot expressed in a In the case of widely separated objects, one can simplify finite numbers of terms. Hence, there is the following the sub-problem by neglecting the contribution of the relation between the eccentric anomaly and the angle
: quadrupole and higher multipole momenta of the bodies to their external gravitational field, thereby approximating the cos
 cos= . equations of orbital motion of two extended bodies by the 1
 cos equations of motion of two point masses located at the Newtonian center of mass of the extended objects. Then the D. Parabolic and Hyperbolic Orbits sub-problem can be exactly solved as shown in the above These solutions can be dealt together. They correspond to Section. The two body problem in GR is more complicated: E
0 which is the condition to obtain unbounded orbits. because of the non-linear hyperbolic structure of Einstein's field equations, one is not sure of the good formulation of Equivalently, one has C
μ / L2 . boundary conditions at infinity, so that the problem is not even well posed [53]. Moreover, since in Einstein's theory The trajectory is the local equations of motion are contained in the

1 gravitational field equations, it is a priori difficult to separate r = l 1+  cos , (2.15) () the problem in two sub-problems, as in the non-relativistic case, where one can compute the gravitational field as a where 
1 . The equal sign corresponds to E =0 . linear functional of the matter distribution independently of Therefore, in order to ensure positivity of r , the polar angle its motion. Furthermore, even when one can achieve such
has to be restricted to the range given by separation and derive some equations of orbital motion for 1+  cos
>0. the two bodies, these equations will a priori not be ordinary differential equations but, because of the finite velocity of This means cos >
1 , i.e.  (
,
) and the trajectory is propagation of gravity, will consist in some kind of retarded integro-differential system [54]. However, all these not closed any more. For , we have r . The   ±

difficulties can be somehow dealt with if one resorts to curve (2.15), with
=1, is aparabola. For
>1, the approximation procedures and breaks the general covariance by selecting special classes of coordinates systems [55]. allowed interval of polar angles is smaller than  (
,
) , 114 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

Two physically different situations, amenable to 1 L r (t), r (t), v (t), v (t) = L + L , (3.1) perturbation treatments, have been considered in the PN ()1 2 1 2 N c2 2 literature: with 1. The problem of two weakly self-gravitating, slowly moving, widely separated fluid bodies which has been 1 1 Gm m L = m v2 m v2 1 2 , (3.2) treated by the so-called PN approximation schemes (for N 1 1 + 2 2 + 2 2 R references see [38, 43, 56, 57]), and 2. The problem of two strongly self-gravitating, widely separated bodies which has been treated by matching a 1 4 1 4 L2 = m1v1 + m2v2 + strong field "internal" approximation scheme in and near the 8 8 objects to a weak field "external" approximations scheme outside the objects. Gm m  2 2 2 M  + 1 2 3(v + v )
7(v v )
N v v
G , (3.3) 2R 1 2 1 2 1 2 R The approach has been pursued both for slowly moving   objects, either BHs [58] or in general strongly self- where we have introduced the instantaneous relative position gravitating objects [59], and for strongly self-gravitating objects moving with arbitrary velocities [60, 61]. In the latter R vector Rrr= 12
and RR=| | while N = . In (3.1) and case, equations of orbital motion were considered in the form R 2 of a retarded-integro-differential system which however (3.2) we used the short notations: vvvv1111
=| |= , could be transformed into ordinary differential equations and vv
= vv for the ordinary Euclidean scalar products, and which, when attention was restricted to slowly moving 12 12 c is the velocity of light. The invariance, at the PN v bodies, were expanded in power series of [61, 62]. When approximation, and modulo an exact time derivative, of L c PN under spatial translations and Lorentz boosts implies, via keeping only the first relativistic corrections to Newton's law Noether's theorem, the conservation of the total linear (first post-Newtonian approximation), it turns out that the momentum of the system: equations of orbital motion of widely separated, slowly moving, strongly self-gravitating objects depend only on two
L
L parameters (the Schwarzschild masses) and are identical to P = PN + PN , PN
v
v the equations of motion of weakly self-gravitating objects 1 2 (when using, in both cases, a coordinate system which is harmonic at lowest order). This is, in fact, a non-trivial and of the relativistic center of mass integral consequence of the structure of Einstein's theory [54]. Then, K = G
tP , in the next subsections, we consider the PN motion including PN PN PN secular and periodic effects at first order approximation and 2 we shall show that the equations of motion can be written in  1 m v 1 Gm m  G = m + 1  1 2 r, a quasi-Newtonian form. PN
 1 2 c2 2 Rc2    A. Relativistic Motion and Conservation Laws the sum is over the two objects [57, 62]. By a Poincaré The relativistic case can be seen as a correction to the transformation it is possible to get a PN center of mass frame where P = K =0. In this frame one has: Newtonian theory of orbits [54, 62]. In GR, the time is PN PN incorporate as a mathematical dimension, so that the four- dimensional rectangular perifocal coordinates are (x, y, z,t) μ μ(m
m )  2 GM  r = R + 1 2 V
R, and the four dimensional polar coordinates are (r,
,,t) . 1 2 2   m1 2M c  R  The first post Newtonian equations of orbital motion of a binary system constrain the evolution in coordinate time t of   μ μ(m1
m2 ) 2 GM the positions r1 and r2 of the two objects. These positions r =
R + V
 R, (3.4) 2 m 2M 2c2 R represent the center of mass in the case of weakly self- 2   gravitating objects (see e.g. [54, 57]) and the center of field dR in the case of strongly self gravitating objects (see [61]). where V = = v1
v2 is the istantaneous relative velocity. They can be derived from Lagrangian which is function of dt The problem of solving the motion of the binary system is the simultaneous position r (t), r (t) , and velocities 1 2 then reduced to the simpler problem of solving the relative

dr dr2 motion in the PN center of mass frame. For the sake of v1(t)= and vt2 ()= in a given harmonic coordinate dt dt completeness, let us write down these equations of motion system, and of two constant parameters, the Schwarzschild derived from (3.1)-(3.2), and where, after variation, the masses of the objects m and m : positions and velocities are replaced by their center of mass 1 2 expressions (3.4): The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 115

dV GM 2 1 1 Gμ M = 4GMN ( + 2)
R 2Nc + L = MW 2 V 2 2 3  ( PN + μ + + dt 2c R 2 2 R +4(NV )V (
2)
3(NV )2 N + 2NV 2 (3 +1) , (3.5) 1  μV V  ) L R W , W . (3.7) + 2 2  +
+  c  m1 m2  where we have introduced a mass parameter μ m m 1 Hence one obtains as a consequence of the equations of the = = 1 2 with 0 . At this point it is
2 

 PN motion: M (m + m ) 4 1 2  worth to notice that in spite of the fact that it is in general 1 L 
d
 1 GM = PN = V 2 incorrect to use, before variation, in a Lagrangian a O  + + μ  R 
R dt
R 2 R consequence, like Eq. (3.4), of the equations of motions, which are obtained only after variation, it turns out that the 1  μ μ   L R, V W , V W , (3.8) relative motion in PN center of mass frame, Eq. (3.5), can be + 2 2  +
+ μc  m m  correctly derived from a Lagrangian obtained by replacing in 1 2  1 the total Lagrangian (divided by μ ) L r ,r ,v ,v the 1 2 μ PN ()1 2 1 2 where in the last bracket we have discarded MW which 2 positions and velocities by their PN center of mass gives no contribution. The first two terms in the (rhs) of Eq. expressions obtained from (3.4) and that moreover it is even (3.8) yield the Newtonian relative motion. We wish to sufficient to use the non-relativistic center of mass evaluate the relativistic corrections to the relative motion: expressions: 
  L  in the PN center of mass frame. Now L2 is a μ 
R  μc2  r = R, 1N m 1 polynomial in the velocities and therefore a polynomial in W , and from Eq. (3.4) one sees that in the PN center of μ r = R,
1 
2N m mass frame W = O . Therefore as does not act on 2  c2  R  
μ W , we see that the contributions coming from W to the rhs v = V , 1N m
1  1 of Eq. (3.8) are of the second PN order O that we shall  c4  μ v = V . (3.6) consistently neglect throughout this work. In other words 2N m 2 one obtains as a consequence of the equations of the PN motion in the PN center of mass frame: The proof goes as follows [54]. Let us introduce the following linear change spatial variables in the PN  1 GM 1  μ μ  V 2 R, V , V  r r  + + 2 L2
 1 2 R 2 R m m Lagrangian L r
r , , : (r ,r )
(R, X ) with  μc  1 2  PN  1 2  1 2    dt dt 
1  = . (m r + m r ) O 4 R = r r and X = 1 1 2 2 , that is:  c  1
2 M This shows that the equations of the relative motion in r = r + X , 1 1N the PN center of mass frame derive from the following r = r + X , Lagrangian: 2 2N dX 1 GM 1  μ μ  which implies (denoting = W ): LR (R,V )= V 2 L R, V , V , PN + + 2 2 
 dt 2 R μc m m  1 2  v = v +W , which happens to be obtained by replacing in the full PN 1 1N Lagrangian, see Eq. (3.7) above, X and W by zero, i. e. the v = v +W . original variables by Eq. (3.6) and by dividing by [54]. 2 2N μ R Expressing The explicit expression of LPN reads:

4  1  R 1 2 GM 1 V L = L r
r ,v ,v +   L r
r ,v ,v , LPN (R,V )= V + + (1
3) + PN N ()1 2 1 2 c2 2 ()1 2 1 2 2 R 8 c2   given by Eq. (3.1)-(3.2) in terms of the new variables one GM  2 2 GM  (3+ )V + (NV )
. (3.9) finds: 2Rc2 R   116 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

The Lagrangian (3.9) was obtained in [63]. The d
H I integration of the equations (3.5) can be done in several = + , (3.13) dt r 2 r 3 different ways. • A standard approach: Lagrange's method of variation where the coefficients ABCDH,,,, , I are the following the osculating elements. polynomials in E and JJ=| | : • The Hamilton-Jacobi equation approach which, takes  3 E  A =2E 1+ (3
1) , advantage of the existence of the PN Lagrangian is the route  2   2 c  which has been taken by Landau and Lifshits [26], who worked out only the secular precession of the periastron.  E  2 B = GM 1+ (7
6) , • Another approach, based on the Maupertuis principle ,  2   c  which reduces the PN problem to a simple auxiliary Newtonian problem. 2 2 2  E  G M C =
J 1+ 2(3
1) + (5
10) , To describe the motion, it is convenient to use the standard  c2  c2 method to solve the non-relativistic two-bodies problem and which consists in exploiting the symmetries of the relative GMJ 2 R R D =(
3 + 8) , Lagrangian L . The invariance L under time 2 PN PN c translations and space rotations implies the existence of four  E  first integrals: H = J 1+ (3
1) ,  c2 
LR
LR E = V  PN  LR and J = R  PN : PN GMJ
V
V I =(2 4) . (3.14) 
2 1 GM 3 V 4 c E = V 2 (1 3 )
+
 2 + 2 R 8 c The relativistic "relative motion", i.e. the solution of Eq. (3.12) can be simply reduced to the integration of auxiliary GM  2 2 GM  (3+ )V + (NV )
, (3.10) non-relativistic radial motion. Indeed let us consider the 2Rc2 R   following change of the radial variable:  1 V 2 GM  D J = R V 1+ (1
3) + (3+ ) . (3.11) r = r + , (3.15) 2 c2 2Rc2 2C   0 It is checked that these quantities coincide respectively
1 2 where C0 is the limit of C when c  0 with (C0 =
J ) .
1 with μ times the total Noether energy and the total Geometrically, the transformation which is expressed in Noether angular momentum of the binary system when polar coordinates by the equation: r
= r+ const ,
 =
, is computed in the PN center of mass frame [64]. Eq. (3.11) called aconchoidal transformation [54]. Taking into account implies that the motion takes place in a coordinate plane,
1  therefore one can introduce polar coordinates R = r and
the fact that D is O and that we can neglect all terms  c2  in the plane (i.e. r = r cos
, r = r sin
, r =0). Then x y z
1  starting from the first integrals (3.10)-(3.11) and using the of order O , we find that replacing Eq. (3.15) in Eqs.  4  2 2  c  2  dr  2  d
 2 d identities: V =   + r   , R
V |= r , (3.1)-(3.3), leads to:  dt   dt  dt 2 dr 3
dr  2B C NV = , we obtain by iteration   = A+ + , (3.16) dt dt r r 2   2
dr  2B C D with = A , (3.12)   + + 2 + 3  dt  r r r BD C = C
. C 0 2 In classical mechanics, Maupertuis' principle is an integral equation that Then, in the case of quasi-elliptical motion (E <0;A <0), determines the path followed by a physical system without specifying the time parameterization of that path. It is a special case of the more generally r is a linear function of cos E , E being an eccentric stated principle of least action. More precisely, it is a formulation of the D equations of motion for a physical system not as differential equations, but anomaly and the same is true of r = r + . We then as an integral equation, using the calculus of variations. 2C 3In these and the following equations we neglect terms of the second PN 0 obtain the PN radial motion in quasi-Newtonian parametric
1  order O . form ( t being a constant of integration):  c4  0   The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 117

nt()=sin,

t0 Et E (3.17)  E r =1 (3 8) , + 
2 t c ra=(1cos),rr
 E (3.18) with  GM  3  r =1+ a c2 4
 . 3/2 r  (
A) t 2  2  n = , B The relativistic angular motion, i.e. the solution of Eq. (3.13) 1/2 can also be simply reduced to the integration of an auxiliary  A  BD  =1 C  , 1 t 2

 B C  non relativistic angular motion. Let us first make, at O   0   c2    A D order, the following conchoidal transformation: ar =
+ , B 2C I 0 r = r
+ , (3.20) 2H  AD 
=1 + 
. which transforms Eq. (3.13) into r  2BC  t  0  d
H The main difference between the relativistic radial = 2 , motion and the non-relativistic one is the appearence of two dt r
eccentricities: the time eccentricity
appearing in the t where r
can be expressed as Kepler equation (3.17) and the relative radial eccentricity
. Using (3.14) we can express a ,,

and n in terms of r
= a
(1

cos E). (3.21) r rrt E and J : Let us note also the relationship between er and et : GM1 E a =1(7), r 

 2 I E 2 c a
= ar
, 2H  2E  5 15 E  =1 1  AI  r + 2 2  +  
 2  = 1 . (3.22) 2 2 
r 
  G M  c   2BH 

1/2 The differential time is given, from Eq. (3.17) by:  G2 M 2  J 2 + (
6) , c2  dt = n
1(1
 cos E)dE.   t Hence we get  2E   7 17   =1 + 1+ 

 t 2 2 2 2  G M   H (1
t cos E) d = dE. na
2 (1

cos E)2 2 2  G M  J 2 + (
2 + 2) , 2 As can be seen from Eq. (3.17) and Eq. (3.22)
and

 c   t 1 differ by only small terms of order . Now if we introduce (
2E)3/2  1 E  c2 n = 1+ (
15) . (3.19) GM 4 c2 any new eccentricity say  also very near
so that we can  
t

1 (t + )  1  It is remarkable that a well known result of the write: = 2 ,  = 
 , with = O t + 
 2  ( + ) 2  c  Newtonian motion is still valid at PN level: both the relative t
semi-major axis ar the mean motion n depend only on the then center of mass energy E . The same is true for the time of 2  ( + )  2
t  2 2 return to periastron period P = . As a consequence we (1
t cos E)(1
 cos E)= 1
cos E
 cos E. n   2   can also express n in term of ar : Therefore if we choose  such that the average of
and
t 1/2 is equal to i.e. =2 we have GM  GM  


 

t n = 1+ (
9 + ). 3 2  ar  2ar c  2 (1
 cos E) 1  1  t = , Let us note also that the relationships between e and e are: + O 4 r t (1

cos E) 1
 cos E  c   118 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis which transforms Eq. (III A) into a Newtonian like angular notation f for the polar angle counted from a periastron and motion equation corrected for the periastron precession i.e.:

H dE 
 d = , f = 0 , na
2 1
 cos E K  which is easily integrated to give We must eliminate E between: r = a (1 cos E), = KA (E), (3.23) r
r 
0   and
being a constant of integration and where for the sake of 0 f = A (E).  simplicity we have introduced the notations:
 1  In order to get, it is convenient to play a new conchoidal 1+ 2 E A (E)=2arctan tan , (3.24) transformation on r writing:    1
 2         r = r a (1
 cos E) + a 1
r . (3.28) and  r  r       H K = . (3.25) From the definition of AE() we have: 1 
2 2 na
(1
 )2 1 2 1 2 

 1
 cos E = = .  1 e A (E) 1 cos f From Eq. (3.22) and (3.19) and the definition of  =2

 +   +  t  we have: Moreover we find from Eq. (3.26) that the radial displacement appearing in Eq. (3.28) is simply  AD AI   AD AI  = 1 = 1  t  +
r  +
BC BH 2BC BH    Gμ  0   0  a 1
r = , r   2c2 then, as shown by straightforward calculations:    so that we find the polar equation of the relative orbit as:  Gμ   =  1+  = 2
r  2  1 2a c  Gμ 
 Gμ  r  r = a
+ r 2 1 cos f 2  2c  + 2c 1    2 2 2  2E 1 15 E 2 G M  This equations means that the relative orbit is the 1+ 1+  
 J
6  , (3.26)  G2 M 2 2 2 c2 c2  conchoid of a precessing ellipse, which means that it is    obtained from an ellipse: r = l(1+ ecos)
1 by a radial and displacemnet r = r
+ const together with an angular J homothetic transformation: = const . Let us finally note K = . (3.27)

 1 that the relative orbit con also be written as: 2 2 2  2 6G M  J
 2 c2 a (1
 )   r = r r , 1+ cos f  As it is clear from Eqs. (3.20) and (3.21), the radial variable r r reaches its successive minima "periastron passages" for with E =0,2
,4
,... The periastron therefore precesses at each GM   turn by the angle  =2
(L 1) , which if J >> f
= f + 2 2 f sin f . c    r  reduces to the well-known result: The conservation laws and the coordinate G2 M 2  1  6
GM  1  transformations which we have obtained here will reveal  =6
+ O = + O . particularly useful to characterize the relativistic orbits, as J 2c2  c4 a (1  )c2  c4 R r we will see below. Contrarily to the usual approach which derives first the B. Relativistic Quasi-Elliptical Orbits orbit by eliminating the time between Eq. (3.12) and (3.13) before working out the motion on the orbit we find the orbit The relativistic motions of each body are obtained by replacing the solutions for the relative motion, by eliminating E between Eq. (3.18) and (3.23)-(3.25). t(E), r(E), (E) , in the PN center of mass formulae Eq. With the aim to simplify the formulae we introduce the
The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 119

(3.4) (see [54]). We see first that the polar angle of the first r = a (1
 cos E), r r object (of mass m1 ) is the same as the relative polar angle r = a (1
e cos E), (3.29) and that the polar angle of the second object (mass m2 ) is r r simply  +
. Therefore it is sufficient to work out the with radial motions. From Eq. (3.4) we have by replacing V 2 in GM  1 E  2GM 2GM GM a = 1
(
7) , the relativistic corrections with + 2E

: r 2E 2 c2 R R a   r 3 m R Gμ(m
m )  R  (
2E)2  1 E  r = 2 + 1 2 1
 n = 1
(
15) . (3.30) M 2mc2  a  GM 4 c2  R   

(and similar results for the second object by exchanging m1 and K, ,e , ,e , a ,e , a given in terms of the total t  r r r r
r
and m2 ) which shows remarkably enough, that r can also be written in a quasi-Newtonian form: angular momentum by unit reduced mass in the center of mass frame, E and J , by Eq. (3.19), (3.27), (3.26), and r = a (1
 cos E), interchange of m and m for ,a . The above equations r r 1 2  rr

are very similar to the standard Newtonian solutions of the with non-relativistic two-body problem. m 2 ar
= ar , C. Relativistic Quasi-Hyperbolic Orbits M The simplest method to obtain the Post-Newtonian   Gm1(m1
m2 E >0  = e 1
, motion in the hyperbolic-like case ( ) consists simply r R 2 in making, in Eqs. (3.29)-(3.30), the analytic continuation  2Mar c  from E <0 to E >0, passing below E =0 in the complex and where as before: 2 E plane and replacing the parameter E by iF ( i =
1). n(t
t )=E
 sin E, The proof that this yields to a correct parametric solution 0 t consists in noticing, on one hand, that K and the various = KA (E). eccentricities are analytic in E , near E =0, and that if one 
0   replaces the parametric solution (3.29)-(3.30) and the

The orbit in space of the first object can be written by corresponding expressions of trrrrr,,,,,,eeaea

in terms using the same method as in the preceding Section for the 2 2
dr   d
 relative orbit, that is: of E and J in   and in , one finds that Eq. dt  dt          r = r a (1
 cos E) + a 1
r . (3.12) and the square of Eq. (3.13) are satisfied identically,  r  r     
1  modulo O , and that the resulting identities are analytic  c4  One finds:   2 in E and E as purely imaginary ones. One finds:    Gm m a 1
r = 1 2 , r 
 2M 2c2 n(t
t )= sinh F
F,   0 t hence we find that the orbits is conchoid of a precessing  1  ellipse with 1 2   + F  
0 = K2arctan  tanh , 2 2 2 1 2   Gm m  1
 Gm m 
r = a 1 2  1 2   r
2 2 + 2 2    2M c   
  2M c 1+  cos 0  L r = a (1
 cos F),   r r Summarizing then gathering our results for the elliptic- r = a (1
 cos F), (3.31) like ( E <0) PN motion in the PN center of mass frame, we r r have: where K, , , , are functions of E and J , as before, t 
r r n(t
t0 )=E
t sin E, but where it has been conveninet to introduce the opposites  1  of analytic continuations of the semi-major axes: 1 2  +  E  
 = K2arctan tan , GM  1 E  0 1  2 
 ar = 1
(
7) ,   2E 2 c2     120 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

m a IV. RELATIVISTIC ORBITS WITH and 1 r and the modulus of the analytic continuation ar
= GRAVITOMAGNETIC CORRECTIONS M of the mean motion: Using the orbital theory developed up to now for 3 3 v 2 relativistic orbit, we have neglected terms of order . (2 M )  1 E  c n = 1
(
15) . GM 4 c2 However, we succeed in explaining, for instance, the   perihelion precession of Mercury. In cases where 3 D. Relativistic Quasi-Parabolic Orbits v v 102

101 , higher order terms like cannot be The quasi-parabolic post-Newtonian motion ( E =0) can c c be obtained as a limit when E
0 . For istance, let us start discarded in order to discuss consistently the problem of from the quasi-elliptic solution in Eq. (3.29) and pose motion (see for example [65, 66]). In this situations, we are

1 dealing with gravitomagnetic corrections. Before discussing 
2E 2 the theory of orbits under the gravitomagnetic effects, let us E =   x. give some insight into gravitomagnetism and derive the G2 M 2   corrected metric. Theoretical and experimental aspects of Taxing now the limit E  0
, holding x fixed, yields gravitomagnetism are discussed in [67, 68]. the following parametric representation of the quasi A remark is in order at this point: any theory combining, parabolic motion: in a consistent way, Newtonian gravity together with Lorentz invariance has to include a gravitomagnetic field generated 1  G2 M 2 1  t t = J 2 (2 2 ) x x3 , (3.32) by the mass-energy currents. This is the case, of course, of
0 2 2 +
 2 +  2G M   c 3  GR: it was shown by Lense and Thirring [69-72], that a rotating mass generates a gravitomagnetic field, which, in turn, causes a precession of planetary orbits. In the J x = 2arctan , (3.33) framework of the linearized weak-field and slow-motion 
0 2 2 1 2 2 1 2 G M 2 2 6G M 2 approximation of GR, the gravitomagnetic effects are (J
6) 2 ) (J
2 ) c c induced by the off-diagonal components of the space-time metric tensor which are proportional to the components of 1  G2 M 2  the matter-energy current density of the source. It is possible r = J 2 + (
6) x + x . (3.34) to take into account two types of mass-energy currents. The 2GM c2    former is induced by the matter source rotation around its center of mass: it generates the intrinsic gravitomagnetic Moreover let us point out that our solutions (for the three field which is closely related to the angular momentum cases E <0, E >0 and E =0) have been written in a (spin) of the rotating body. The latter is due to the 2 translational motion of the source: it is responsible of the 2
GM  suitable form when J >6  . However the validity of extrinsic gravitomagnetic field [73]. Let us now discuss the c gravitomagnetic effects in order to see how they affect the   2 orbits. 2 GM  our solutions can be extended to the range J
6  by A. Gravitomagnetic Effects c   first replacing in the solutions for the angular motion , the Starting from the Einstein field equations in the weak field approximation, one obtain the gravitoelectromagnetic second equation of (3.29) and (3.31), and considering the equations and then the corrections on the metric4 [65]. The Eqs. (3.32)-(3.34), the function arctan by cot (at the price weak field approximation can be set as of a simple modifiation of
) and then by making an 0 g (x)=
+ h (x), h (x) << 1, (4.1) analytic continuation in J . One ends up with an angular μ μ μ μ motion expressed by an argcoth which can also be where
is the Minkowski metric tensor and h (x) << 1 approximatively replaced by its asymptotic behaviour for μ μ
1 is a small deviation from it [39]. large arguments: argcoth( X )
. The case of purely radial X The stress-energy tensor for perfect - fluid matter is given motion ( J =0) is also obtained by taking the limit J
0 by (at EF, or respectively x fixed). Finally a parametric μ 2 μ  μ representation of the general post-Newtonian motion in an T = p + c u u
pg , () arbitrary (post-Newtonian harmonic) coordinate system is obtained from our preceding center of mass solution by which, in the weak field approximation p
c2 , is
applying a general transformation of the Poincaré group a a b a x
= Lb x + T [54]. 4Notation: Latin indices run from 1 to 3, while Greek indices run from 0 to 3; the flat space-time metric tensor is . μ =diag (1,


1, 1, 1) The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 121

T 00

c2 , T 0 j

cv j , T ij

viv j . where the dot indicates the differentiation with respect to the affine parameter. In order to put in evidence the 4 gravitomagnetic contributions, let us explicitly calculate the From the Einstein field equations G =(8
G / c )T , one μ μ Christoffel symbols at lower orders. By some finds straightforward calculations, one gets

0 2 8
G =0 (4.11) h = , (4.2)
00  00 2  c 0 1
 = (4.12) 2 8
G 0 j 2 j h = , (4.3) c x  ij 2 ij 
c i j 0 2 
V
V  2 16
G l  =   + (4.13)  h =   v , (4.4) ij 3 j i 0 j 2 jl c 
x
x  c 2 k 1
 where
is the standard Laplacian operator defined on the = (4.14) 00 2 k flat spacetime. To achieve Eqs. (4.2)-(4.4), the harmonic c
x condition k j k 2 
V
V  gμ
 =0,  =   (4.15) μ 0 j c3 x j xk 

 has been used. k 1 
 k
 k
  By integrating Eqs. (4.2)-(4.4), one obtains  =   +    (4.16) ij c2 x j i xi j xk ij 


2 h00 =
, (4.5) In the approximation which we are going to consider, we c2 2 j 3 are retaining terms up to the orders
/ c and V / c . It is 2 important to point out that we are discarding terms like hij =
ij , (4.6) c2 ( / c4 ) / xk , (V j / c5 ) / xk , (/)cV5 kj / x, 





kji6 4 (/)VcV

/ x and of higher orders. Our aim is to show h =
V l . (4.7) 0 j 3 jl that, in several cases like in tight binary stars, it is not correct c to discard higher order terms in vc/ since physically The metric is determined by the gravitational Newtonian interesting effects could come out. The geodesic equations potential up to c
3 corrections are then

 3 (x)= G d x , (4.8) 2 j    2 d t 2
 dt dx
x x' c c  2 + 2 j d c
x d d and by the vector potential V l , 2 
V m
V m  dxi dx j l  im + jm =0, (4.17) l v 3 c3 x j xi d d V = G
d x . (4.9) 

 x  x' for the time component, and l given by the matter current density
v of the moving 2 bodies. This last potential gives rise to the gravitomagnetic d 2 xk 1
  dt  1
 dxi dx j + c +  corrections. d 2 c2 x j d c2 xk ij d d 
 
From Eqs. (4.1) and (4.5)-(4.9), the metric tensor in terms of Newton and gravitomagnetic potentials is 2
 dxl dxk 4 
V k
V m  dt dxi c =0, (4.18)  2 l + 3 j  jm k c
x d d c 
x
x d d l 2  2 2 2 8ljV j ds =1 c dt cdtdx for the spatial components.  + 2
3 +  c  c 2 2 In the case of a null-geodesic, it results ds = d
=0. Eq. (4.10) gives, up to the order c
3 ,  2 i j 1 dx dx . (4.10)

2 lj l  c  4V l  2 cdt = dx + 1
dl , (4.19) c3 c2 euclid From Eq. (4.10) it is possible to construct a variational   principle from which the geodesic equation follows. Then where dl 2 =
dxi dx j is the Euclidean length interval. we can derive the orbital equations. As standard, we have euclid ij Squaring Eq. (4.19) and keeping terms up to order c
3 , one x

 +
 x
μ x
 =0, μ finds 122 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

l The gravitomagnetic term is the second one in Eq. (73) 2 2  4 2 8V l c dt =1
dleuclid + dx dleuclid . (4.20) and it is usually discarded since considered not relevant. This  c2  c3 is not true if v / c is quite large as in the cases of tight binary systems or point masses approaching to black holes. Inserting Eq. (4.20) into Eq. (4.18), one gets, for the spatial components, Our task is now to achieve explicitly the trajectories, in particular the orbits, corrected by such effects. 2 d2 xklk22
dl
 dx dx euclid 22+ kl  2 + B. Gravitomagnetically Corrected Orbits dcx

ddd cx Orbits with gravitomagnetic effects can be obtained 4 
V k
V m  dl dx j starting from the classical Newtonian theory and then euclid =0. (4.21) 3 j  jm k c 
x
x  d d correcting it by successive relativistic terms. Starting from the above considerations (see Sec. I, and III) we can see how Such an equation can be seen as a differential equation gravitomagnetic corrections affect the problem of orbits. for dxk / d
which is the tangent 3-vector to the trajectory. Essentially, they act as a further vc/ correction leading to 3 On the other hand, Eq. (4.21) can be expressed in terms of take into account terms up to c
, as shown above. Let us l considered as a parameter. In fact, for null geodesics start from the line element (4.10) which can be written in euclid spherical coordinates. Here we assume the motion of point- and taking into account the lowest order in v / c , d
is like bodies and then we can work in the simplified proportional to dleuclid . From Eq. (4.17) multiplied for GM l l hypothesis  =
and V =
v . It is  2  r 1+
 , we have c2   2  2 2  2 ds =1 cdt 1  + 2 

2  d  dt 2 dt 4 dxi   c   c  V m =0, + 2 
4 im d  d c d c d  2 2 2 2 2 2 dr + r d
+ r sin
d  and then     i 8 dt 2 4 m dx
cdt cos + sin cos + sin dr 1+ 
 V =1, (4.22) 3  () d c2 c4 im d c   8 where, as standard, we have defined the affine parameter so + cdt cos cos + sin
sinrd 3  () that the integration constant is equal to 1 [39]. Substituting c Eq. (4.19) into Eq. (4.22), at lowest order in v / c , we find 8 + cdt sin cos
sin rd. dleuclid 3  () =1. (4.23) c cd
In the weak field regime, the spatial 3-vector, tangent to a As in the Newtonian and relativistic cases, from the line given trajectory, can be expressed as element (4.26), we can construct the Lagrangian dxk cdx k 2 2 2 = . (4.24) L =1+ t

1
r
+ r 2
2 + r 2 
2  d dl 2  2  sin
euclid c c  

Through the definition 8
t
cos + sin cos + sin r
dxk 3  () ek = , c dl euclid 8 + t
cos cos + sin
sinr
Eq. (4.21) becomes 3  () c k de 2
 2
 l k +  e e + 8 2 k 2 l + t
sin cos
sin r
. (4.26) dl c
x c
x 3  () euclid c

k m 4 
V
V  j e =0,  2
 + 3  j  jm k Being, =1, one can multiply both members for 1 . L  + 2  c 
x
x  c   which can be expressed in a vector form as In the planar motion condition  =
/2, we obtain

2 de 2 4 2  2 2 2 L  =
 
e(e  )+ e ( V). (4.25) E 1 1 r 2   3  
 + 2 
2 
+ 2  dleuclid c c  c  c  r   The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 123

8E Our aim is to show how gravitomagnetic effects modify
 cos + sin r

cos
sin 
 3 ()() the orbital motion and what the parameters determining the c stability of the problem are. As we will see, the energy and the mass, essentially, determine the stability. Beside the  2
 =1 + , standard periastron precession of GR, a nutation effect is c2   genuinely induced by gravitomagnetism and stability greatly depends on it. A fundamental issue for this study is to 2 R achieve the orbital phase space portrait. and then, being = s (where R is the Schwarzschild 2
s c r In Fig. (2), the results for a given value of nutation 1 angular velocity with a time span of 10000 steps is shown. It radius) and u = it is is interesting to see that, by increasing the initial nutation r angular velocity, being fixed all the other initial conditions, E 2
h2 1
R2u2 u2 + u2 + we get curves with decreasing frequencies for rt
() and

rt(). ()s () This fact is relevant to have an insight on the orbital motion stability (see Fig. 7). The effect of gravitomagnetic terms are 4R uE S ()cos + sin u + ()cos
sin u2  taken into account, in Fig. (3-4), showing the basic orbits c   (left) and the orbit with the associated velocity field in false colors (right). From a rapid inspection of the right panel, it is =1
R u . ()S clear the sudden changes of velocity direction induced by the gravitomagnetic effects. By deriving such an equation, it is easy to show that, if the relativistic and gravitomagnetic terms are discarded, the Newtonian theory is recovered, being R u u = s .

+ 2 2L This result probes the self-consistency of the problem. However, it is nothing else but a particular case since we have assumed the planar motion. This planarity condition does not hold in general if gravitomagnetic corrections are taken into account. >From the above Lagrangian (4.26), it is straightforward to derive the equations of motion

1 2 r

= [crc2 + GM

2 + sin

2 r 2 + cr rc2 + 2GM ()() () Fig (2). Plots along the panel lines of: rt() (upper left),phase |
4GMt
((cos(cos + sin)
sin)
+ portrait of rt() versus rt
() (bottom left), rt
() (upper right) and 2 2 rt() (bottom right) for a star of 1M . The examples we are sin(cos
sin)
r + cGMr

cGMt
], (4.27)


) showing were obtained solving the system for the following 2 parameters and initial conditions: μ
1M , E = 0.95 ,
=0, 

=
ccot rc2 + 2GM 

r 2
0 r 2 rc3 2GMc () 1 1 1 ()+
,

,
and r = and r =20 . The 0 =
0 = 0 0 =
r
0
0
0 μ 2 10 10 100 stiffness is evident from the trend of rt
() and

rt(). +r
2GM csc(sin
cos)t
+ cr rc2 + GM 
, (4.28) ()() To show the orbital velocity field, a rotation and a 1


= ccos
r 2 rc2 + 2GM sin

2 projection of the orbits along the axes of maximal energy can 2 3 () r ()rc + 2GMc be performed. In other words, by a Singular Value Decomposition of the de-trended positions and velocities, the eigenvectors corresponding to the largest eigenvalues can be +r
4GM (cos(cos + sin)
sin)t
+ selected, and, of course, those representing the highest ( energy components (see Figs. 3-4). 2
2cr rc + GM 
, (4.29) The above differential equations for the parametric ()) orbital motion are non-linear and with time-varying corresponding to the spatial components of the geodesic Eq. coefficients. In order to have a well-posed Cauchy problem, (4.21). The time component

t is not necessary for the we have to define: discussion of orbital motion. Being the Lagrangian (4.26) • the initial and final boundary condition problems; L =1 it is easy to achieve a first integral for r
which is a natural constrain equation related to the energy. • the stability and the dynamical equilibrium of solutions. 124 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

Fig. (5). Breaking points examples: on the left panel, the first four orbits in the phase plane are shown: the red one is labelled I, the Fig. (3). Plots of basic orbits (left) The initial values are: ; green is II, the black is III and the fourth is IV. As it is possible to r
E = 0.95 in mass units; r =20 in Schwarzschild radii;
= ; see, the orbits in the phase plane are not closed and they do not n 0 
10 overlap at the orbital closure points; we have called this feature 
breaking points. In this dynamical situation, a small perturbation

= . 10 can lead the system to a transition to the chaos as prescribed by the Kolmogorov-Arnold-Moser (KAM) theorem [25]. On the right panel, it is shown the initial orbit with the initial (square) and final (circles) points marked in black.

Fig. (4). Plots of basic orbits with the associated velocity field. The arrows indicate the instantaneous velocities. The initial values are: ; r
E = 0.95 in mass units; r =20 in Schwarzschild radii;
= ; n 0 
10 Fig. (6). In this figure it is shown the initial orbit with the initial 
(squares) and final(circles) points marked in black.

= . 10 mass, the energy, the orbital radius, the initial values of We can start by solving the Cauchy problem, as in the and the angular precession and nutation velocities r,,


classical case, for the initial condition putting r
=0,

=0, and

respectively. We have empirically assumed initial
conditions on r , and
.

=0 and  = and the result we get is that the orbit is not


2 The trials can be organized in two series, i.e. constant planar being 


0 . In this case, we are compelled to solve mass and energy variation and constant energy and mass numerically the system of second order differential equations variation. and to treat carefully the initial conditions, taking into account the high non-linearity of the system. A similar • In the first class of trials, we assume the mass equal to M = discussion, but for different problems, can be found in [74, 1M and the energy E (in mass units) varying step by 75]. n step. The initial orbital radius r0 can be changed, A series of numerical trials on the orbital parameters can according to the step in energy: this allows to find out be done in order to get an empirical insight on the orbit numerically the dynamical equilibrium of the orbit. We stability. The parameters involved in this analysis are the have also chosen, as varying parameters, the ratios of the The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 125

find a guess on the equilibrium of the initial radius, followed by trials and errors procedure. • In the second class of trials, we have assumed the variation of the initial orbital radius for different values of mass at a constant energy value equal to E = 0.95 in mass units. n r
With this conditions, we assume

= and assume that 10

takes the two values 1/2 and 1/10. One can approach the problem also considering the mass parameterization, at a given fixed energy, to have an insight of the effect of mass variation on the initial conditions. The masses have been varied between 0.5 and 20 Solar masses and the distances have been found to vary according to the two 3rd-order polynomial functions, according to the different values of

with respect to the mass (for details see [65]). In summary, the numerical calculations, if optimized, allow to put in evidence the specific contributions of gravitomagnetic corrections on orbital motion. In particular, specific contributions due to nutation and precession emerge when higher order terms in v / c are considered. The conclusion of this part of the review is that orbits are highly characterized by the velocity regime of the moving bodies. The order of the parameter vc/ determines the global shape of the trajectories. Our task is now to show how the motion of sources is related to the features of emitted GWs, in particular to their production and to the profile of waves.

PART II

Production and Signature of Gravitational Waves Fig (7). Plots of orbits with various energy values. For each value The first derivation of gravitational radiation in GR is of energy, four plots are shown: the first on the left column is the due to Einstein. His initial calculation [76] was "marred by orbit, with the orbital velocity field in false colors. The color scale an error in calculation" (Einstein's words), and was corrected goes from blue to red in increasing velocity. The second on the left in 1918 [77] (albeit with an overall factor of two error). column is the orbit with a different nutation angular velocity. On Modulo a somewhat convoluted history (discussed in great the right column, the phase portraits r
= r
(r(t)) are shown. Energy detail by Kennefick [78]) owing (largely) to the difficulties varies from 0.3 to 0.4, in mass units. The stability of the system is of analyzing radiation in a nonlinear theory, Einstein's final highly sensitive either to very small variation of r or to variation result stands today as the leading-order "quadrupole 0 formula" for gravitational wave emission. This formula plays on the initial conditions on both precession and nutation angular a role in gravity theory analogous to the dipole formula for velocities: it is sufficient a variation of few percent on r to induce 0 electromagnetic radiation, showing GWs arise from system instability. accelerated masses exactly as electromagnetic waves arise from accelerated charges. The quadrupole formula tells us precession angular velocity

to the radial angular that GWs are difficult to produce – very large masses velocity r
and the ratio of the nutation angular velocity moving at relativistic speeds are needed. This follows from

and the precession angular velocity

. The initial the weakness of the gravitational interaction. A consequence of this is that it is extremely unlikely there will ever be an condition on
has been assumed to be
=0 and the 0 interesting laboratory source of GWs. The only objects
massive and relativistic enough to generate detectable GWs initial condition on
has been  = . For M =1 (in 0 2 are astrophysical. Indeed, experimental confirmation of the existence of GWs has come from the study of binary neutron

1 r
Solar masses) , = and 
=
, can be found out star systems – the variation of the mass quadrupole in such
2  10 systems is large enough that GW emission changes the two different empirical linear equations, according to the system's characteristics on a timescale short enough to be different values of

,
. One obtains a rough guess of the observed. Intuitively, it is clear that measuring these waves must be difficult – the weakness of the gravitational initial distance r = r (E ) around which it is possible to 0 0 n interaction ensures that the response of any detector to 126 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis gravitational waves is very small. Nonetheless, technology where hh= μ is the trace of the metric perturbation, and has brought us to the point where detectors are now μ 2 2 beginning to set interesting upper limits on GWs from some
=

 =  
is the wave operator. Contracting once  t sources [79-82]. The first direct detection could more, we find the curvature scalar: be,hopefully, not too far in the future. R = Rμ =

μ h 
h μ () μ V. GRAVITATIONAL WAVES IN LINEARIZED and finally build the Einstein tensor: GRAVITY 1 G = R R The most natural starting point for any discussion of μ μ
μ GWs is linearized gravity [38, 43, 83]. Linearized gravity is 2 an adequate approximation to GR when the spacetime 1 metric, may be treated as deviating only slightly from a flat = h  h h h

 μ +

μ  
μ 
μ
 metric, : 2 (
μ g =
+ h ,||h ||
1.Here ||h || means ``the 

 h + 
h . μ μ μ μ μ
μ   μ ) magnitude of a typical non-zero component of h ''. Note μ
This expression is a bit unwieldy. Somewhat remarkably, that the condition || h ||
1 requires both the gravitational it can be cleaned up significantly by changing notation: μ
5 field to be weak , and in addition constrains the coordinate rather than working with the metric perturbation hμ
, we use system to be approximately Cartesian. We will refer to h 1 μ
the trace-reversed perturbation h = h
 h . (Notice as the metric perturbation; as we will see, it encapsulates μ μ 2 μ GWs, but contains additional, non-radiative degrees of that h μ =
h , hence the name “trace reversed''.) Replacing freedom as well. In linearized gravity, the smallness of the μ 1 perturbation means that we only keep terms which are linear h with h +
h in Eq. (5.2) and expanding, we find μ
μ 2 μ in hμ
– higher order terms are discarded. As a consequence, that all terms with the trace h are canceled. What remains is indices are raised and lowered using the flat metric
μ . The metric perturbation h transforms as a tensor under Lorentz 1 μ
G =

h  +

h 
h  

 h  . (5.2) μ ()  μ μ μ μ μ   transformations, but not under general coordinate 2 transformations. This expression can be simplified further by choosing an We now compute all the quantities which are needed to appropriate coordinate system, or gauge. Gauge describe linearized gravity. The components of the affine transformations in general relativity are just coordinate connection (Christoffel coefficients) are given by transformations. A general infinitesimal coordinate transformation can be written as xa'=xμ μ , where 1 +
μ = μ
h +
h 
h  ()      μ (x
) is an arbitrary infinitesimal vector field. This 2  1 transformation changes the metric via =.
hhμμμ+

h 2 ()    μ h '=h 2 , (5.3) μ μ 
(μ ) Here
μ means the partial derivative
/
x . Since we use to raise and lower indices, spatial indices can be written
μ so that the trace-reversed metric becomes either in the "up" position or the "down" position without x 1 changing the value of a quantity: f = f . Raising or h '=h '
 h x μ μ μ t 2 lowering a time index, by contrast, switches sign: ff=
t . = h  2
 +
 . The Riemann tensor we construct in linearized theory is then μ ( μ) μ  given by A class of gauges that are commonly used in studies of Rμ =
μ 
μ radiation are those satisfying the Lorenz gauge condition      μ 1
hμ =0. (5.4) =

hμ +

μ h 

μ h 

hμ . (5.1) 2 ()          (Note the close analogy to Lorentz gauge 6 From this, we construct the Ricci tensor

1 6 Fairly recently, it has become widely recognized that this gauge was in fact R = R =

h +

h 
h 

h , μ μ ()  μ μ  μ μ  invented by Ludwig Lorenz, rather than by Hendrik Lorentz. The inclusion 2 of the "t" seems most likely due to confusion between the similar names; see Ref. [84] for detailed discussion. Following the practice of Griffiths ([85], p. 421), we bow to the weight of historical usage in order to avoid any possible 5We will work in geometrized coordinates putting cG==1 confusion. The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 127 in electromagnetic theory,
μ A =0, where A is the flat (for our purposes, h
0 as r 
). Equivalently, we μ μ μ potential vector.) Suppose that our metric perturbation is not specialize to the space of homogeneous, asymptotically flat solutions of the linearized Einstein Eq. (5.6). For such in Lorentz gauge. What properties must
μ satisfy in order spacetimes one can, along with choosing Lorentz gauge, to impose Lorentz gauge? Our goal is to find a new metric further specialize the gauge to make the metric perturbation h such that
μ h =0: μ
μ be purely spatial μ μ μ  h = h =0 (5.8)
h '=
h 

 
 +

 00 0i μ μ  μ    and traceless =
μ h 
 . μ  h = h i =0. (5.9) i Any metric perturbation hμ
can therefore be put into a Lorentz gauge by making an infinitesimal coordinate The Lorentz gauge condition (5.4) then implies that the transformation that satisfies spatial metric perturbation is transverse:

μ
h =0.
 =
h . (5.5) i ij  μ One can always find solutions to the wave equation (5.5), This is called the transverse-traceless gauge, or TT thus achieving Lorentz gauge. The amount of gauge freedom gauge. A metric perturbation that has been put into TT gauge has now been reduced will be written hTT . Since it is traceless, there is no μ
The amount of gauge freedom has now been reduced TT TT distinction between h and h . from 4 freely specifiable functions of 4 variables to 4 μ
μ
functions of 4 variables that satisfy the homogeneous wave The conditions (5.8) and (5.9) comprise 5 constraints on equation

=0, or, equivalently, to 8 freely specifiable the metric, while the residual gauge freedom in Lorentz functions of 3 variables on an initial data hypersurface. gauge is parameterized by 4 functions that satisfy the wave equation. It is nevertheless possible to satisfy these Applying the Lorentz gauge condition (5.4) to the conditions, essentially because the metric perturbation expression (5.2) for the Einstein tensor, we find that all but satisfies the linearized vacuum Einstein equation. When the one term vanishes: TT gauge conditions are satisfied, the gauge is completely 1 fixed. G = h . μ

μ 2 One might wonder why we would choose the TT gauge. Thus, in Lorentz gauges, the Einstein tensor simply It is certainly not necessary; however, it is extremely reduces to the wave operator acting on the trace reversed convenient, since the TT gauge conditions completely fix all metric perturbation (up to a factor
1/2). The linearized the local gauge freedom. The metric perturbation hTT μ
Einstein equation is therefore therefore contains only physical, non-gauge information
h = 16
T ; (5.6) about the radiation. In the TT gauge there is a close relation μ μ between the metric perturbation and the linearized Riemann in vacuum, this reduces to tensor R [which is invariant under the local gauge μ

h =0. (5.7) transformations (5.3) by Eq. (5.1)], namely μ
Just as in electromagnetism, the equation (5.6) admits a 1

TT class of homogeneous solutions which are superpositions of Ri0 j0 =
hij . 2 plane waves: In a globally vacuum spacetime, all non-zero components h (x,t)=Re d 3kA (k)ei(kxt) . μ
μ of the Riemann tensor can be obtained from R via i0 j0 Here,
=| k | . The complex coefficients A (k) depend on Riemann's symmetries and the Bianchi identity. In a more μ
general spacetime, there will be components that are not the wavevector k but are independent of x and t . They are μ related to radiation. subject to the constraint kAμ
=0 (which follows from the μ Transverse traceless gauge also exhibits the fact that Lorentz gauge condition), with k =(
, k) , but are gravitational waves have two polarization components. For otherwise arbitrary. These solutions are the gravitational example, consider a GW which propagates in the z waves. TTTT direction: hhtzij=() ij
is a valid solution to the wave A. Transverse Traceless (TT) Gauge in Globally Vacuum equation
hTT =0. The Lorentz condition
hTT =0 implies Spacetimes ij z zj that hTT (t
z)=constant . This constant must be zero to We now specialize to globally vacuum spacetimes in zj which T =0 everywhere, and which are asymptotically satisfy the condition that h
0 as r 
. The only non- μ
ab 128 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

TT TT TT TT TT where
is the proper time as measured by an observer zero components of hij are then hxx , hxy , hyx , and hyy . travelling along the geodesic. By writing the derivatives in Symmetry and the tracefree condition (5.9) further mandate the above geodesic equation in terms of coordinate time t that only two of these are independent: rather than proper time , and by combining the (i.e
μ = t hTT =
hTT  h (t
z); 0 coordinate) equation with the spatial, μ = j (i.e. spatial xx yy + coordinates) equations, we obtain an equation for the hTT = hTT  h (t
z). coordinate acceleration: xy yx  2 i The quantities and are the two independent d x i i j i j k h+ h
= ( 2 v v v ) 2
00 + 0 j +  jk + waveforms of the GW (see Fig. 8, 9) [38, 86]. dt vi (
0 + 2
0 v j +
0 v j vk ), (6.1) 00 0 j jk

i i where v = dx / dt is the coordinate. Let us now specialize to linearized theory, with the non- flat part of our metric dominated by a GW in TT gauge. Further, let us specialize to non-relativistic motion for our i test body. This implies that v
1 , and to a good approximation we can neglect the velocity dependent terms in Eq. (6.1):

d 2 xi +
i =0. Fig. (8). We show how point particles along a ring move as a result dt 2 00 of the interaction with a GW propagating in the direction perpendicular to the plane of the ring. This figure refers to a wave In linearized theory and TT gauge, with + polarization. i 0 1 TT TT  00 = i0 = 2
t hj0 
j h00 =0, 2 () since hTT =0. Hence, we find that d 2 xi / dt 2 =0. 00 This result could mean that the GW has no effect.This is not true since it just tells us that, in TT gauge, the coordinate location of a slowly moving, freely falling (here in after FF) body is unaffected by the GWs. In essence, the coordinates move with the waves. This result illustrates why, in GR, it is important to focus upon coordinate-invariant observables (a naive interpretation Fig. (9). We show how point particles along a ring move as a result of the above result would be that freely falling bodies are not of the interaction with a GW propagating in the direction influenced by GWs). In fact the GWs cause the proper perpendicular to the plane of the ring. This figure refers to a wave separation between two FF particles to oscillate, even if the with
polarization. coordinate separation is constant. Consider two spatial FF particles, located at z =0, and separated on the x axis by a To illustrate the effect of GWs on free falling (FF) coordinate distance L . Consider a GW in TT gauge that particles, we consider a ring of point particles initially at rest c with respect to a FF frame attached to the center of the ring, propagates down the z axis, hTT (t, z) . The proper distance as shown in Fig. (8, 9). μ
L between the two particles in the presence of the GW is VI. INTERACTION OF GRAVITATIONAL WAVES given by

WITH A DETECTOR L L L = cdx g = cdx 1+ hTT (t, z =0) The usual notion of “gravitational force” disappears in
0 xx
0 xx GR, replaced instead by the idea that freely falling bodies L  1  follow geodesics in spacetime. Given a spacetime metric g
cdx 1+ hTT (t, z =0) = μ

0  2 xx    and a set of spacetime coordinates xμ , geodesic trajectories
 are given by the equation 1 TT = Lc 1+ hxx (t, z =0) . (6.2)  2  d 2 xμ dx dx μ =0, 2 +
 Notice that we use the fact that the coordinate location of d d d each particle is fixed in TT gauge. In a gauge in which the The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 129 particles move with respect to the coordinates, the limits of detectors with L >
(e.g. the space based detector LISA) integration would have to vary. Eq. (6.2) tells us that the the analysis here is
not valid and other techniques must be proper separation of the two particles oscillates with a used to analyze the detector. fractional length change given by
L / L A convenient coordinate system to analyze the geodesic deviation equation (6.4) is the local proper reference frame
L 1 TT
hxx (t, z =0). (6.3) of the observer who travels along the first geodesic. This L 2 coordinate system is defined by the requirements Although we used TT gauge to perform this calculation, zi ( )=0, g (t,0)= ,
μ (t,0)= 0, (6.5) the result is gauge independent; we will derive it in a μ μ  different gauge momentarily. Notice that hTT acts as a xx which imply that the metric has the form strain, a fractional length change. The magnitude h of a  x2  wave is often referred to as the “wave strain”. The proper ds2 =
dt 2 + dx2 + O . (6.6) R2 distance we have calculated here is a particularly important   quantity since it directly relates to the accumulated phase which is measured by laser interferometric GW observatories Here R is the radius of curvature of spacetime, given by . The “extra” phase
 accumulated by a photon that travels R
2
|| R || . It also follows from the gauge conditions μ down and back the arm of a laser interferometer in the (6.5) that proper time
and coordinate time t coincide
i presence of a GW is  =4
L /  , where
is the along the first geodesic, that u = and that Lμ =(0,L ) .
t photon's wavelength and
L is the distance the end mirror moves relative to the beam splitter7. We now give a different Consider now the proper distance between the two i ii derivation of the fractional length change (6.3) based on the geodesics, which are located at x =0 and xLt=(). From concept of geodesic deviation. Consider a geodesic in the metric (6.6) we see that this proper distance is just spacetime given by xμ = zμ (
) , where
is the proper time, 2 2 ||=LLLii, up to fractional corrections of order L / R . with four velocity uμ ( )=dzμ / d . Suppose we have a

For a GW of amplitude h and wavelength
we have nearby geodesic xμ ( )=zμ ( ) Lμ ( ) , where Lμ ( ) is R
2 h / 2 , so the fractional errors are hL2 / 2 . (Notice

+




small. We can regard the coordinate displacement Lμ as a that R
L / h the wave's curvature scale R is much larger
vector L = Lμ
on the first geodesic; this is valid to first than the lengthscale L characterizing its variations.) Since μ
we are restricting attention to detectors with L

, these order in L . Without loss of generality, we can make the fractional errors are much smaller than the fractional connecting vector be purely spatial: Luμ =0. Spacetime μ distance changes
h caused by the GW. Therefore, we can curvature causes the separation vector to change with time, simply identify | L | as the proper separation. the geodesics will move further apart or closer together, with an acceleration given by the geodesic deviation equation We now evaluate the geodesic deviation equation (6.4) in the local proper reference frame coordinates. From the
u  (u  Lμ )=
Rμ [z( )]u Lu ; (6.4) conditions (6.5) it follows that we can replace the covariant   
time derivative operator uμ
with

/(t ). Using u =
see, e.g., Ref. [91]. This equation is valid to linear order in μ t Lμ ; fractional corrections to this equation will scale as and LLμ = (0,i ) , we get LL/ , where L is the lengthscale over which the curvature varies. d 2 Li (t) = R (t,0)Lj (t). (6.7) 2
i0 j0 For application to GW detectors, the shortest lengthscale dt L is the wavelength
of the GWs. Thus, the geodesic Note that the key quantity entering into the equation, deviation equation will have fractional corrections of order R , is gauge invariant in linearized theory, so we can use L /
. For ground-based detectors L < a few km, while i0 j0

> 3000km ; thus the approximation
will be valid. For any convenient coordinate system to evaluate it. Using the
expression (V A) for the Riemann tensor in terms of the TT TT 7 gauge metric perturbation h we find This description of the phase shift only holds if L

, so that the metric ij perturbation does not change value very much during a light travel time. 2 i 2 TT This condition will be violated in the high frequency regime for space-based d L 1 d hij j GW detectors; a more careful analysis of the phase shift is needed in this = L . 2 2 2 case [87]. Furtheremore fractional corrections of order L /
can be impor- dt dt tant also for ground-based detectors. An important case regards relic GWs Integrating this equation using Ltiii()= L+
Lt () with where the spectrum is continuous and in the high frequency portion of the 0 band of ground-based detectors is involved. In this case, as it has been care- |
L |
| L0 | gives fully explained in [88] fractional corrections of order L/lambda are impor- tant for ground-based detectors too. On the other hand, such fractional cor- 1 rections of order L /
represent the so called "magnetic" component of the i TT j
L (t)= hij (t)L0 . (6.8) signal, as it has been carefully explained in [89]. 2 130 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

This equation is ideal to analyze an interferometric GW vicinity of the detector will also be present. From Eq. (6.7) detector. We choose Cartesian coordinates such that the we see that the quantity that is actually measured by interferometer's two arms lie along the x and y axes, with interferometric detectors is the space-time-space-time or the beam splitter at the origin. For concreteness, let us electric-type piece R of the Riemann tensor (or more i0 j0 imagine that the GW propagates down the z -axis. Then, as precisely the time-varying piece of this within the frequency discussed in Sec. V A, the only non-zero components of the band of the detector). From the general expression of TT TT TTTT metric perturbation are hxx =
hyy = h and hhhxy== yx
, Riemann tensor (see [38]), we see that R contains + i0 j0 where h (t z) and h (t z) are the two polarization TT +

contributions from both h describing GWs, and also ij components. We take the ends of one of the interferometer's additional terms describing the time-varying near zone two arms as defining the two nearby geodesics; the first gravitational fields. There is no way for the detector to geodesic is defined by the beam splitter at x =0, the second separate these two contributions, and the time-varying near by the end-mirror. From Eq. (6.8), we then find that the zone gravitational fields produced by motions of bedrock, distances L =| L | of the arms end from the beam splitter air, human bodies, and tumbleweeds can all contribute to the vary with time as output of the detector and act as sources of noise [92-94].
L 1 x = h , A. The Generation of Gravitational Waves + L 2 GWs are generated by the matter source term on the right hand side of the linearized Einstein equation L 1 y =
h . L 2 +
h = 16
T , (6.9) μ μ (Here the subscripts x and y denote the two different arms, cf. Eq. (5.6) (presented here in Lorentz gauge). In this not the components of a vector). These distance variations section we will compute the leading order contribution to the are then measured via laser interferometry. Notice that the spatial components of the metric perturbation for a source GW (which is typically a sinusoidally varying function) acts whose internal motions are slow compared to the speed of tidally, squeezing along one axis and stretching along the light (``slow-motion sources''). We will then compute the TT other. In this configuration, the detector is sensitive only to piece of the metric perturbation to obtain the standard the + polarization of the GW. The
polarization acts quadrupole formula for the emitted radiation. similarly, except that it squeezes and stretches along a set of axes that are rotated with respect to the x and y axes by Eq. (6.9) can be solved by using a Green's function. A 45
. The force lines corresponding to the two different wave equation with source generically takes the form polarizations are illustrated in Fig. (10). f (t, x)=s(t, x),
where ftx(, ) is the radiative field, depending on time t and position x , and stx(, ) is a source function. Green's function Gtxt(, ;
, x ') is the field which arises due to a delta function source; it tells how much field is generated at the “field point” (,tx ) per unit source at the “source point'' : (t
, x') G(t, x;t, x') = (t
t)(x
x') .
The field which arises from our actual source is then given by integrating Green's function against stx(, ):

3 f (t, x)= dtd xG (t, x;t, x')s(t, x') .
Fig (10). Lines of force associated to the + (left panel) and
(right panel) polarizations. The Green's function associated with the wave operator
is very well known (see, e.g., [95]): Of course, we don't expect nature to provide GWs that so perfectly align with our detectors. In general, we will need to (t [t | x  x'|/c]) account for the detector's antenna pattern, meaning that we G(t, x;t, x') =  . 4 | x x'| will be sensitive to some weighted combination of the two
 polarizations, with the weights depending upon the location of a source on the sky, and the relative orientation of the The quantity t
| x
x'|/c is the retarded time; it takes source, the frequency and the detector [44, 88-90]. into account the lag associated with the propagation of Finally, in our analysis so far of detection we have information from events at x to position x' . The speed of assumed that the only contribution to the metric perturbation light c has been restored here to emphasize the causal nature is the GW contribution. However, in reality time-varying of this Green's function; we set it back to unity in what near zone gravitational fields produced by sources in the follows. The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 131

Applying this result to Eq. (6.9), we find Next, manipulate the right-hand side of Eq. (6.11); multiplying by xxij, we obtain: T (t | x x'|,x') 3 μ   h (t, x)=4 d x . kl i j kl i j ik j kj i ij ab
 | x x'|
k
lT x x =
k
l T x x  2
k T x + T x + 2T .  ()() Projected transverse and traceless, as already mentioned, This identity is easily verified by expanding the derivatives jj the radiative degrees of freedom are contained entirely in the and applying the identity
iix =  . We thus have spatial part of the metric. First, consider the spatial part of the metric:
2 T 00 xi x j =

T kl xi x j  2
T ik x j + T kj xi + 2T ij . t ()k l ()k () ij 3 T (t | x  x'|,x') h (t, x)=4 d x . This yields ij
| x  x'| 4 4 1 d 3xT = d 3x 2 T 00 x'i x'j   ij  
t + We have raised indices on the right-hand side, using the r r 2 () rule that the position of spatial indices in linearized theory is irrelevant. ik 'j kj 'i 1 kl 'i 'j  +
T x + T x 

T x x  We now evaluate this quantity at large distances from the k ()2 k l ()  source. This allows us to replace the factor |'|xx
in the 2 denominator with r =| x | . The corresponding fractional = d 3x 2 T 00 x'i x'j  
t r () errors scale as
L / r , where L is the size of the source; these errors can be neglected. We also make the same 2 2
3 00 'i 'j = d xT x x replacement in the time argument of T : 2  ij r
t

2 T (t | x  x'|,x')
T (t  r, x'). 2
3 'i 'j ij ij = d x x x . 2  r
t Using the formula | x
x'|=r
ni x'i + O(1 / r) where In going from the first to the second line, we used the fact i i n = x / r , we see that the fractional errors generated by the that the second and third terms under the integral are replacement (VI A) scale as L /
, where
is the timescale divergences. Using Gauss's theorem, they can thus be recast over which the system is changing. This quantity is just the as surface integrals; taking the surface outside the source, velocity of internal motions of the source (in units with their contribution is zero. In going from the second to the c =1), and is therefore small compared to one by our third line, we used the fact that the integration domain is not time dependent, so we can take the derivatives out of the assumption. These replacements give 00 integral. Finally, we used the fact that T is the mass 4 density
. Defining the second moment Q of the mass h (t, x)= d 3xTij (t r, x') , (6.10) ij ij
  distribution via r Q (t)= d 3x (t, x')x'i x'j , (6.12) which is the first term in a multipolar expansion of the ij
 radiation field. and combining Eqs. (6.10) and (6.12) we get Eq. (6.10) almost gives us the quadrupole formula that 2 describes GW emission (at leading order). To get the 2 d Q (t
r) h (t, x)= ij . (6.13) remaining part there, we need to manipulate this equation a ij 2 r dt bit. The stress-energy tensor must be conserved, which When we subtract the trace from Q , we obtain the means
T μ =0 in linearized theory. Breaking this up into ij μ quadrupole momentum tensor: time and space components, we have the conditions 1 Q = Q
 Q, Q = Q .
T 00 +
T 0i =0, ij ij ij ii t i 3 To complete the derivation, we must project out the non-
T 0i +
T ij =0. t j TT pieces of the right-hand side of Eq. (6.13). Since we are working to leading order in 1/r , at each field point x , this From this, it follows that operation reduces to algebraically projecting the tensor perpendicularly to the local direction of propagation
2T 00 =

T kl . (6.11) t k l n = x / r , and subtracting off the trace. It is useful to introduce the projection tensor, Multiplying both sides of this equation by xxij, we first P = 
n n . manipulate the left-hand side: ij ij i j 2 00 i j 2 00 i j This tensor eliminates vector components parallel to n ,
t T x x =
t T x x . () leaving only transverse components. Thus, 132 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

h T = h P P shortcoming of the above linearized-gravity derivation of the ij kl ik jl quadrupole formula was first pointed out by Eddington. is a transverse tensor. Finally, we remove the trace; what However, it is very straightforward to extend the derivation remains is to encompass systems with non-negligible self gravity. In full GR, we define the quantity h μ
via 1 hTT = h P P
P P h . (6.12) ij kl ik jl 2 ij kl kl μ μ μ
gg = 
h ,

Substituting Eq. (6.13) into (6.14), we obtain our final μ quadrupole formula: where   diag(
1,1,1,1) . When gravity is weak, this μ
2 definition coincides with our previous definition of h as a TT 2 d Q (t
r)  1  h (t, x)= kl P (n)P (n)
P (n)P (n), trace-reversed metric perturbation. We impose the harmonic ij r dt 2 ik jl 2 kl ij   gauge condition or
( ggμ )=
h μ =0. (6.19) μ μ 2G In this gauge, the Einstein equation can be written as hTT (t, x)= Q (t r)P . (6.15) ij 4 kl
ijkl rc
h μ = 16
(T μ + tμ ), (6.20) flat One can now search for wave solutions of (6.9) from a system of masses undergoing arbitrary motions, and then where
 μ

is the flat-spacetime wave operator, and flat μ  obtain the power radiated. The result, assuming the source tμ
is a pseudotensor that is constructed from h μ
. Taking a dimensions very small with respect to the wavelengths, coordinate divergence of this equation and using the gauge dE (quadrupole approximation [26]), is that the power condition (6.19), stress-energy conservation can be written d
radiated in a solid angle
is
(T μ + tμ )=0. (6.21) μ 2 3 Eqs. (6.19)- (6.20) and (6.21) are precisely the same dE G  d Q  =  ij  (6.16) equations as are used in the linearized-gravity derivation of d 8
c5  dt3  the quadrupole formula, except for the fact that the stress   μ
μ
μ
If one sums (6.16) over the two allowed polarizations, one energy tensor T is replaced by T + t . Therefore the obtains derivation of the last subsection carries over, with the

modification that the formula (6.12) for Qij is replaced by  3 3 3 3 dE G d Qij d Qij d Qij d Qkj =   2n n + 3 00 00 'i 'j
5 3 3 i 3 k 3 Q (t)= d xT (t, x') + t (t, x')x x . pol d 8c  dt dt dt dt ij
   00 2 In this equation the term t describes gravitational 2 3 3  1  d 3Q  1  d Q  d 3Q d Q binding energy, roughly speaking. For systems with weak
 ii + n n ij + ii n n jk (6.17) 2  dt3 2  i j dt3 dt3 j k dt3 gravity, this term is negligible in comparison with the term     00  T describing the rest-masses of the bodies. Therefore the where nˆ is the unit vector in the radiation direction. The quadrupole formula (6.15) and the original definition (6.12) of Q continue to apply to the more general situation total radiation rate is obtained by integrating (6.17) over all ij directions of emission; the result is considered here. GQ(3)Q(3)ij dE ij C. Dimensional Analysis F GW = = (6.18)
5 dt 45c The rough form of the leading GW field that we just where the index (3) represents the number of differentiations derived, Eq. (6.15), can be deduced using simple physical with respect to time, the symbol <> indicates that the arguments. First, we define some moments of the mass quantity is averaged over several wavelengths. distribution. The zeroth moment is just the mass itself: M  d 3x = M . B. Extension to Sources with Non-Negligible Self Gravity 0
More accurately, this is the total mass-energy of the source. Concerning our derivation of the quadrupole formula Next, we define the dipole moment: (6.15) we assumed the validity of the linearized Einstein equations. In particular, the derivation is not applicable to M x d 3x = ML . 1 
 i i systems with weak (Newtonian) gravity whose dynamics are 8 L is a vector with the dimension of length; it describes the dominated by self-gravity, such as binary star systems . This i displacement of the center of mass from the origin we chose.

As such, M1 is clearly not a very meaningful quantity – we 8Stress energy conservation in linearized gravity, μ , forces all
Tμ =0 can change its value simply by choosing a different origin. bodies to move on geodesics of the Minkowski metric. The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 133

If our mass distribution exhibits internal motion, then G d 2 M h
2 . moment of the mass current, j =
v , are also important. 4 2 i i c dt r The first momentum is the spin angular momentum: There is no conservation principle that allows us to reject S  v x  d 3x = S . this term. This is the quadrupole formula we derived earlier, 1
j k ijk i up to numerical factors. Finally, we look at the second momentum of the mass distribution: In “normal” units, the prefactor of this formula turns out to be G / c4 – a small number divided by a very big number. 3 M   x x d x = ML 2
i j ij In order to generate interesting amounts of GWs, the where L is a tensor with the dimension length squared. variation quadrupole momentum must be enormous. The ij only interesting sources of GWs will be those which have Using dimensional analysis and simple physical very large masses undergoing extremely rapid variation; arguments, it is simple to see that the first moment that can even in this case, the strain we expect from typical sources is contribute to GW emission is M . Consider first M . We tiny. The smallness of GWs reflects the fact that gravity is 2 0 want to combine M with the distance to our source, r , in the weakest of the fundamental interactions. 0 such a way as to produce a dimensionless wavestrain h . The D. Numerical Estimates only way to do this (bearing in mind that the strain should Consider a binary star system, with stars of mass m and fall off as 1/r , and restoring factors of G and c ) is to put 1 m2 in a circular orbit with separation R . The quadrupole G M h 0 . moment is given by
2 c r  1 2  Conservation of mass-energy tells us that M0 for an Qij = μ xi x j
R ij  , (6.23)  3  isolated source cannot vary dynamically. This h cannot be radiative; it corresponds to a Newtonian potential, rather where x is the relative displacement, with ||=xR. We use than a GW. the center-of-mass reference frame, and choose the Let us consider now the momentum M . In order to get coordinate axes so that the binary lies in the xy plane, so 1 the right dimensions,we must take one time derivative: x = x = Rcos
t , y = x = Rsin
t , z = x =0. Let us 1 2 3 G d M further choose to evaluate the field on the z axis, so that n h
1 . points in the z -direction. The projection operators in Eq. 3 dt r c (6.15) then simply serve to remove the zj components of the The extra factor of c converts the dimension of the time tensor. Bearing this in mind, the quadrupole formula (6.15) derivative to space, so that the whole expression is yields dimensionless. Think carefully about the derivative of M : 1 TT 2Qij dM1 d 3 3 h = . = x d x = v d x = P . ij
 i
 i i r dt dt This is the total momentum of our source. Our guess for the The quadrupole moment tensor is form of a wave corresponding to M1 becomes  1  G P 2
t  cos
t sin
t 0 h
. (6.22)  cos  3  3  c r  2 1  Also this formula cannot describe a GW. The momentum 2 cos
t sin
t cos
t  0 Q = μR  3  ; of an isolated source must be conserved. By boosting into a ij   1 different Lorentz frame, we can always set P =0. Terms  00  like this can only be gauge artifacts; they do not correspond  3   to radiation. Indeed, terms like (6.22) appear in the metric of   a moving BH, and correspond to the relative velocity of the BH and the observer [96]. its second derivative is

Dimensional analysis tells us that radiation from S1 must take the form  cos 2
t sin 2
t 0   sin 2 t cos 2 t 0 G d S1 2 2 

 h
. Qij = 2
μR . c4 dt r  000   Conservation of angular momentum tells us that the total   spin of an isolated system cannot change, so we must reject TT also this term. Finally, we examine M : The magnitude h of a typical non-zero component of h is 2 ij 134 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

2 2 2/3 2/3 5/3 4μ
R 4μ M
96  M  h = = . f
=
8/3 f 11/3 . (6.27) r r GW  3  GW 5  c  9 We used Kepler's 3rd law to replace R with powers of Introducing the time to coalescence  = t
t , and coal the orbital frequency
and the total mass M = m + m . 1 2 integrating Eq. (6.28), we get The combination of masses here, μ M 2/3 , appears quite often 5/8 3/8 1.21M  1sec  in studies of GW emission from binaries; it motivates the f 130  Hz, (6.28) GW
    definition of the chirp mass:  M  


3/5 2/5 M = μ M . (6.24) where 1.21M is the chirp mass of a NS-NS binary. Eq.
For the purpose of numerical estimate, we will take the (6.28) predicts, e.g. coalescence times of members of the binary to have equal masses, so that 3
17min,2sec,1msec , for f
10,100,10 Hz. Using the μ =/4M : GW above equations, it is straightforward to compute the relation M 5/3
2/3 h = . between the radial separation and the GW frequency. We r find Finally, we insert numbers corresponding to plausible 1/3 2/3 sources:
M 
100Hz  R
300    km. (6.29) 5/3 2/3 2.8M f       GW  21 M 1hour  1kiloparsec  h
10
      2 M  P r Finally, a useful quantity is the number of GW cycles,        defined by  5/3 2/3
22 M 0.01sec  100 Megaparsecs  t 
10       . 1 fin 1 fin  2.8M  P r N = (t)dt = d. (6.30)        GW  t  

in
in  The first line corresponds roughly to the mass, distance Assuming


, we get fin in and orbital period ( P =2
/  ) expected for the many close
5/3
5/3 binary white dwarf systems in our G alaxy. Such binaries are  M   f  N 104   in . (6.31) so common that they are likely to be a confusion limited GW
    1.21M 10Hz  source of GWs for space-based detectors, acting in some    cases as an effective source of noise. The second line VII. GRAVITATIONAL WAVES FROM SOURCES IN contains typical parameter values for binary neutron stars NEWTONIAN MOTION that are on the verge of spiralling together and merging. Such waves are targets for the ground-based detectors that We have now all the ingredient to characterize have recently begun operations. The tiny magnitude of these gravitational radiation with respect to the motion of the waves illustrates why detecting GWs is so difficult. The sources, i.e. with respect to different types of stellar emission of GWs costs energy and to compensate for the loss encounters. Let us start with the Newtonian cases. With the above formalism, it is possible to estimate the amount of of energy, the radial separation R between the two bodies energy emitted in the form of GWs from a system of massive must decrease. We shall now derive how the orbital objects interacting among them [34, 35]. Considering the frequency and GW frequency change in time, using quadrupole components for two bodies interacting in a Newtonian dynamics and the balance equation Newtonian gravitational field, we have:

dE 2 2 2 orbit Q = μr (3cos  sin 
1) , =
P. (6.25) xx dt Q = μr 2 (3sin2  sin2 
1) , At Newtonian order, E =
m m /(2R) . Thus, yy orbit 1 2 R
= 2/3(R
)(

/
2 ) . As long as

/
2  1, the radial 1 velocity is smaller than the tangential velocity and the Q = r 2μ(3cos 2 +1) , zz 2 binary's motion is well approximated by an adiabatic (7.1) sequence of quasi-circular orbits. Eq. (6.25) implies that the 3 orbital frequency varies as Q = Q = r 2μ( cos sin 2) , xz zx 2 5/3

96 GM
 =   , (6.26) 3
2 5  c3  Q = Q = r 2μ( sin 2 sin) , yz zy 2 and the GW frequency fGW =2
,   2 3 2 Qxy = Qyx = r μ  sin  sin 2 , 9 32 2 In units with G =1, and for circular orbits of radius R , RM
= .   The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 135 where the masses m and m have polar coordinates The total energy emitted in the form of gravitational 1 2 radiation, during the interaction, is : {r cos
cos, r cos
sin,r sin
} with i =1,2. We will i i i
dE work in the equatorial plane (  =
/2). The origin of the E = dt .  0 dt motions is taken at the center of mass. Such components can be differentiated with respect to time, as in Eq. (6.18), in From Eq. (2.4), we can adopt the angle
as a suitable order to derive the amount of gravitational radiation in the integration variable. In this case, the energy emitted for various Newtonian orbital motions.
<
<
is 1 2 3 A. Gravitational Wave Luminosity from Circular and G  E( , )= 2 f () d , Elliptical Orbits 1 2 5 5
 45c l 1 The first case we are going to consider is that of closed and the total energy can be determined from the previous circular and elliptical orbits. Using Eq. (2.14), let us derive relation in the limits 1
0 and 
2  . Thus, one has the angular velocity equation 4 3 2 G
(m1 + m2 ) μ 2 E = F() Gl(m + m )( cos
+1) 5 5 = 1 2 l c
2 l where F()
depends on the initial conditions only and it is given by and then, from Eqs. (7.1), the third derivatives of quadrupolar components for the elliptical orbits are: 13824 +102448
2 + 59412
4 + 2549
6 F(
)=(). d 3Q 2880 xx = (24 cos (9 cos 2 ) 11))sin 3
 +  +  dt In other words, the gravitational wave luminosity strictly depends on the configuration and kinematics of the binary d 3Q system. yy =
(24 cos +(13+ 9cos 2)) sin ) dt3 B. Gravitational Wave Luminosity from Parabolic and d 3Q Hyperbolic Orbits zz = 2 sin 3
  dt In this case, we use Eq. (2.15) and Eq. (7.1) to calculate the quadrupolar formula for parabolic and hyperbolic orbits. d 3Q The angular velocity is xy = (24 cos (11 9cos 2 )) sin ) 3
 + +   dt = l 2 L( cos 1)2 ,
  + where and the quadrupolar derivatives are

3/2 2 3 Gl(m1 + m2 )) μ( cos +1) d Q = . xx
4 =
(24 cos +(9 cos 2 +11))sin, l 3 dt

Being 3 d Qyy 3 =
(24 cos +(13+ 9cos 2)) sin ), G 3 Q(3)Q(3)ij = (m + m )3 μ2 (1+ cos
)4 dt ij l5  1 2 3 d Qzz 2 =
2 sin, 415 + 3(8cos
+ 3 cos 2
) dt3 ( 3 2 d Q 3 (72 cos
+(70 + 27 cos 2
))) sin
 xy = ( cos 1)2 (5 cos 8cos 2 3 cos3 ),  3
   +   +  +   dt 2 the total power radiated is given by where 3 dE G = l 4 L3 ( cos 1)2 . = f (
),
μ   + dt 45c5l5 The radiated power is given by where dE G 2 =
 dt 120c5 3 2 4 f ()=(m1 + m2 ) μ (1+ cos)
2 [314 + (1152cos(
+187 cos 2
2 (415 + 3(8cos + 3 cos 2)

3(80cos3 + 30 cos 4 + 48cos5 + 9 cos6)) (72 cos (70 27 cos 2 ))) sin 2 .
+ +


192cos 4 + 576], 136 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis then 2G G(m + m )μ(13 cos +12cos 2 +(4 + 3cos3)) h11 =
1 2 , Rc4 2l dE Gl8 L6μ2 =
f (), dt 5 120c 2G G(m + m )μ(17 cos +12cos 2 +(8 + 3cos3)) h22 = 1 2 , where f ( ) , in this case, is Rc4 2l

2 f () = (314 + (1152cos( +187 cos 2
3(80cos3 2G G(m + m )μ(13 cos +12cos 2 +(4 + 3cos3)) h12 = h21 = 1 2 ,
4 Rc 2l +30 cos 4
+ 48cos5
+ 9 cos6
)) +
so that the expected strain amplitude h
(h2 + h2 + 2h2 )1/2 11 22 12
192cos 4 + 576). turns out to be Then using Eq. (6.18), the total energy emitted in the G3 (m + m )μ2 form of gravitational radiation, during the interaction as a h = 1 2 4 2
function of
, is given by c Rl 1 2 E( , )= d (3(13 cos
+12cos 2
+(4 + 3cos3
))
1 2  5   480c 1 2 2 (Gl8 L6
12716 + 24276 4 + 34768 2 + 4608 μ2 ), +(17 cos
+12cos 2
+(8 + 3cos3
)) ) , () which, as before, strictly depends on the initial conditions of and the total energy can be determined from the previous the stellar encounter. A remark is in order at this point. A relation in the limits 
1  and 
2  in the parabolic monochromatic gravitational wave has, at most, two case. Thus, one has independent degrees of freedom. In fact, in the TT gauge, we have h = h h and h = h h (see e.g. [97]). As an + 11 + 22
12 + 21 (Gl8 L6
μ2 E = F( ) , example, the amplitude of gravitational wave is sketched in   5  480c Fig. (11) for a stellar encounter , in Newtonian motion, close to the Galactic Center. The adopted initial parameters are where F()
depends on the initial conditions only and it is typical of a close impact and are assumed to be b =1 AU for given by
1 the impact factor and v0 = 200 Km s for the initial F(
) = 1271
6 + 24276
4 + 34768
2 + 4608 . velocity, respectively. Here, we have fixed ()m = m = 1.4M . The impact parameter is defined as 1 2
In the hyperbolic case, we have that the total energy is L = bv where L is the angular momentum and v the 3
3
incoming velocity. We have chosen a typical velocity of a determined in the limits 1  and 2  , i.e. star in the galaxy and we are considering, essentially, 4 4 Gl8 L6μ2 E = F( ) ,
 5  201600c where F()
depends on the initial conditions only and is given by

F( ) = [315 (1271 6 24276 4 34768 2 4608) 
 +  +  + +

16 [ [ (926704 2 7 (3319 2 32632 2 +   
 


55200)) 383460] 352128 2]] . +
+ As above, the gravitational wave luminosity strictly depends on the configuration and kinematics of the binary system. Fig. (11). The gravitational wave-forms from elliptical orbits shown as function of the polar angle
. We have fixed C. Gravitational Wave Amplitude from Elliptical Orbits m = m = 1.4M . m is considered at rest while m is moving Beside luminosity, we can characterize also the GW 1 2
2 1 with initial velocity v = 200 Km s
1 and an impact parameter amplitude starting from the motion of sources. In the case of 0 a binary system and a single amplitude component , it is b =1 AU. The distance of the GW source is assumed to be R =8 straightforward to show that kpc and the eccentricity is
= 0.2,0.5,0.7. The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 137 compact objects with masses comparable to the Chandrasekhar limit (
1.4M ) . This choice is motivated by  the fact that ground-based experiments like VIRGO or LIGO expect to detect typical GW emissions from the dynamics of these objects or from binary systems composed by them (see e.g. [43]).

D. Gravitational Wave Amplitude from Parabolic and Hyperbolic Orbits The single components of amplitude for a parabolic and hyperbolic orbits are

2 2 11 Gl L μ h =
(13 cos +12cos 2 +(4 + 3cos3)) , Rc4

2 2 22 Gl L μ h = (17 cos +12cos 2 +(8 + 3cos3)) , Rc4 Fig. (12). The gravitational wave-forms for a parabolic encounter as a function of the polar angle
. As above, m = m = 1.4M and 2 2 1 2
12 21 3Gl L μ h = h = (4 cos (cos 2 3))sin , is considered at rest. is moving with initial velocity
4  +  +  m2 m1 Rc Km s
1 with an impact parameter b =1 AU. The v0 = 200 and then the expected strain amplitude is distance of the GW source is assumed at R =8 kpc. The 4 4 2 eccentricity is
=1. 2l L μ 4 3 2 h = (10 9 cos3 59 cos 2 4
 +   +   c R

1 +59 2 + 47 2 +108  cos
+ 36)2 , () which, as before, strictly depends on the initial conditions of the stellar encounter. We note that the gravitational wave amplitude has the same analytical expression for both cases and differs only for the value of
which is
=1 if the motion is parabolic and the polar angle range is ( , ) ,   

>1 while it is
and   (
,
) for hyperbolic orbits. In these cases, we have non-returning objects. The amplitude of the gravitational wave is sketched in Figs. (12 and 13) for stellar encounters close to the Galactic Center. As above, we consider a close impact and assume 1 b =1 AU cm, v = 200 Km s
and m = m = 1.4M . In 0 1 2
summary, we can say that also in the case of Newtonian Fig. (13). The gravitational wave-forms for hyperbolic encounters motion of sources, the orbital features characterize GW as function of the polar angle
. As above, we have fixed luminosities and amplitudes. m = m = 1.4M . m is considered at rest while m is moving 1 2
2 1 VIII. GRAVITATIONAL WAVES FROM SOURCES
1 with initial velocity v = 200 Km s and an impact parameter IN RELATIVISTIC MOTION 0 b =1 AU. The distance of the source is assumed at R =8 kpc. The It is straighforward to extend the above considerations to eccentricity is assumed with the values
= 1.2,1.5,1.7 . orbital motions containing Post
Newtonian corrections. It is clear that GW luminosity and amplitude are strictly A. Inspiralling Waveform Including Post-Newtonian v Corrections dependent on the parameter ( ) considered at various order c As we have shown in the above section the PN method of approximation and, as discussed above, the global feature involves an expansion around the Newtonian limit keeping of orbits fully characterize the gravitational emission. Now terms of higher order in the small parameter [8, 98, 99] we study how the waveforms depend on the dynamics of 2 binary and colliding systems and how relativistic corrections v2
h T 0i T ij 

h
0

. (8.1) modulate the features of gravitational radiation. c2 μ
h T 00 T 00 i 138 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

In order to be able to determine the dynamics of binary 16447322263 1712 16 2  systems with a precision acceptable for detection, it has been 3PN =   E +
+ 139708800 105 3 necessary to compute the force determining the motion of   the two bodies and the amplitude of the gravitational  56198689 451 2  radiation with a precision going beyond the quadrupole  +
 + 217728 48 formula. For nonspinning BHs, the two-body equations of   motion and the GW flux are currently known through 3.5PN 541 5605 856 order [100]. Specifically if we restrict the discussion to  2
 3
log 16v2 , (8.9) 896 2592 105   circular orbits, as Eq. (6.26) shows, there exists a natural adiabatic parameter

/
2  O[(v / c)5 ] . Higher-order PN

 4415 358675 91495 2  corrections to Eq. (6.26) have been computed [100, 101],  =  +  + 
. (8.10) 3.5PN 4032 6048 1512 yielding the general equation:   7 We denote L = μ X
V the Newtonian angular 
96 5/3 k/3 = v v (8.2) 2  
(k/2)PN   5 momentum (with X and V , as above, the two-body center- k=0 of-mass radial separation and relative velocity), and 1/3 where Gc=1= and where we define v
( M) . The Lˆ = L/|L |; S =
m2 Sˆ and S =
m2 Sˆ are the spins of  1 1 1 1 2 2 2 2 PN-order is given by which is, up to k =7,
(k/2)PN the two bodies (with Sˆ unit vectors, and 0<
<1 for 1,2 1,2 BHs) and
=1, (8.3) 0PN  S S 
=0, (8.4) S  S + S ,
M  2  1 . (8.11) 0.5PN 1 2 m m  2 1  743 11 (8.5)
1PN =   , Finally, m = m
m and
= 0.577… is Euler's 336 4 1 2 E constant.  47 S 25 m    =4
+  L  L , (8.6) It is instructive to compute the relative contribution of the 1.5PN 3 M 2 4 M M 2  PN terms to the total number of GW cycles accumulating in the frequency band of LIGO/VIRGO. In Table II, we list the 34103 13661 59
= +  +  2  figures obtained by plugging Eq. (8.2) into Eq. (6.30) [8]. As 2PN 18144 2016 18 final frequency, we use the innermost stable circular orbit

(ISCO) of a point particle in Schwarzschild BH 1 ISCO 247(Sˆ Sˆ ) 721(Lˆ Sˆ )(Lˆ Sˆ ), (8.7) [ f
4400 / ( M / M ) Hz]. 12  1  2
 1  2  GW  48 B. The Full Waveform: Inspiral, Merger and Ring-Down 1  31811 = (4159 15876 ) 2.5PN  + 
+  + After the two BHs merge, the system settles down to a 672  1008 Kerr BH and emits quasi-normal modes (QNMs), [102, 103]. This phase is commonly known as the ring-down (RD) 5039 S  473 1231 m    L +
+  L , (8.8) phase. Since the QNMs have complex frequencies totally 84 M 2  84 56 M M 2 determined by the BH's mass and spin, the RD waveform is a 

Table 2. Post-Newtonian Contributions to the Number of GW Cycles Accumulated from  =
10Hz to
=
ISCO =1/(63/2 M ) for in fin Binaries Detectable by LIGO and VIRGO. We Denote
= Sˆ  Lˆ and
= Sˆ  Sˆ i i 1 2

(10+10)M
(1.4+1.4) M
Newtonian 601 16034

1PN +59.3 +441

1.5PN -51.4+16.0
1
1+16.0
2
2 -211+5.7
1
1+65.7
2
2

2PN +4.1-3.3
1
2
1
2+1.1

1
2 +9.9-8.0
1
2
1
2+2.8

1
2 2.5PN -7.1+5.5
1
1+5.5
2
2 -11.7+9.0
1
1+9.0
2
2

3PN +2.2 +2.6

3.5PN -0.8 -0.9 The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 139

Fig. (14). We sketch the curvature potential as function of the * tortoise coordinate r associated to metric perturbations of a Schwarzschild BH.

Fig (16). GW signal from an equal-mass nonspinning BH binary as predicted at 2.5PN order by Buonanno and Damour (2000) in Ref. [19]. The merger is assumed almost instantaneous and one QNM is included. Today, with the results in NR, we are in the position of assessing the closeness of analytic to numerical waveforms for inspiral, merger and RD phases. In Fig. (16), we show some first-order comparisons between the EOB-analytic and NR waveforms [107] (see also Ref. [108]). Similar results for the inspiral phase but using PN theory [100, 101] Fig (15). The potential peaks at the last unstable orbit for a (without resummation) at 3.5PN order are given in Refs. massless particle (the light ring). Ingoing modes propagate toward [107, 108]. So far, the agreement is qualitatively good, but the BH horizon, whereas outgoing modes propagate away from the more accurate simulations, starting with the BHs farther source. apart, are needed to draw robust conclusions. superposition of damped sinusoidals. The inspiral and RD Those comparisons are suggesting that it should be waveforms can be computed analytically. What about the possible to design purely analytic templates with the full merger? Since the nonlinearities dominate, the merger would numerics used to guide the patching together of the inspiral be described at best and utterly through numerical and RD waveforms. This is an important avenue to template simulations of Einstein equations. However, before construction as eventually hundreds of thousands of numerical relativity (NR) results became available, some waveform templates may be needed to extract the signal analytic approaches were proposed. In the test-mass limit, from the noise, an impossible demand for NR alone.

1, Refs. [103, 104] realized a long time ago that the basic physical reason underlying the presence of a universal merger signal was that when a test particle falls below 3M (the unstable light storage ring of Schwarzschild), the GW that it generates is strongly filtered by the curvature potential barrier centered around it (see Fig. 14). For the equal-mass case
=1/4, Price and Pullin [105] proposed the so-called close-limit approximation, which consists in switching from the two-body description to the one-body description (perturbed-BH) close to the light-ring location. Based on these observations, the effective-one-body (EOB) resummation scheme [19] provided a first example of full waveform by (i) resumming the PN Hamiltonian, (ii) modeling the merger as a very short (instantaneous) phase and (iii) matching the end of the plunge (around the light- ring) with the RD phase (see Ref. [106] where similar ideas were developed also in NR). The matching was initially done using only the least damped QNM whose mass and spin were determined by the binary BH energy and angular momentum Fig (17). GW signal from an equal-mass BH binary with a small at the end of the plunge. An example of full waveform is spin
=
= 0.06 obtained in full GR by Pretorius [107]. given in Fig. (14, 17). 1 2 140 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

IX. GRAVITATIONAL WAVES WITH problem discussing in detail the dynamics of such a GRAVITOMAGNETIC CORRECTIONS phenomenon which could greatly improve the statistics of possible GW sources. In this section, we are going to study the evolution of compact binary systems, formed through the capture of a Besides, we give estimates of the distributions of these moving (stellar) mass m by the gravitational field, whose sources and their parameters. It is worth noticing that the source is a massive MBH of mass M where m
M . One captures occur when objects, in the dense stellar cusp expects that small compact objects (1÷ 20M ) from the surrounding a galactic MBH, undergo a close encounter, so
that the trajectory becomes tight enough that orbital decay surrounding stellar population will be captured by these through emission of GWs dominates the subsequent black holes following many-body scattering interactions at a evolution. According to Refs. [74, 75]), for a typical capture, relatively high rate [109]. It is well known that the capture of the initial orbital eccentricity is extremely large (typically stellar-mass compact objects by massive MBHs could
6
3 constitute, potentially, a very important target for LISA [110, 1
e
10
10 ) and the initial pericenter distance very 111]. However, dynamics has to be carefully discussed in small ( r
8
100M , where M is the MBH mass [115]. p order to consider and select all effects coming from standard The subsequent orbital evolution may (very roughly) be stellar mass objects inspiralling over MBHs. divided into three stages. In the first and longest stage the In the first part of this review, we have shown that, in the orbit is extremely eccentric, and GWs are emitted in short relativistic weak field approximation, when considering “pulses” during pericenter passages. These GW pulses higher order corrections to the equations of motion, slowly remove energy and angular momentum from the gravitomagnetic effects in the theory of orbits, can be system, and the orbit gradually shrinks and circularizes. particularly significant, leading also to chaotic behaviors in 3 8 After
10
10 years (depending on the two masses and the transient regime dividing stable from unstable the initial eccentricity) the evolution enters its second stage, trajectories. Generally, such contributions are discarded where the orbit is sufficiently circular: the emission can be since they are considered too small. However, in a more viewed as continuous. Finally, as the object reaches the last accurate analysis, this is not true and gravitomagnetic stable orbit, the adiabatic inspiral transits to a direct plunge, corrections could give peculiar characterization of dynamics and the GW signal cuts off. Radiation reaction quickly [65, 88-90]. circularizes the orbit over the inspiral phase; however, initial According to these effects, orbits remain rather eccentric eccentricities are large enough that a substantial fraction of until the final plunge, and display both extreme relativistic captures will maintain high eccentricity until the final perihelion precession and Lense-Thirring [69, 112, 113] plunge. It has been estimated [75] that about half of the precession of the orbital plane due to the spin of MBH, as captures will plunge with eccentricity e > 0.2 . While well as orbital decay. In [114], it is illustrated how the individually-resolvable captures will mostly be detectable measured GW-waveforms can effectively map out the during the last
1
100 yrs of the second stage (depending spacetime geometry close to the MBH. In [32, 33], the on the stellar mass m and the MBH mass), radiation emitted classical orbital motion (without relativistic corrections in during the first stage will contribute significantly to the the motion of the binary system) has been studied in the confusion background. As we shall see, the above scenario is extreme mass ratio limit m
M , assuming the stellar system density and richness as fundamental parameters. The heavily modified since the gravitomagnetic effects play a conclusions have been that crucial role in modifying the orbital shapes that are far from being simply circular or elliptic and no longer closed. • the GW-waveforms have been characterized by the orbital motion (in particular, closed or open orbits give rise A. Gravitational Waves Amplitude Considering Orbits to very different GW-production and waveform shapes); with Gravitomagnetic Corrections • in rich and dense stellar clusters, a large production of Direct signatures of gravitational radiation are given by GWs can be expected, so that these systems could be very interesting for the above mentioned ground-based and space GW-amplitudes and waveforms. In other words, the detectors; identification of a GW signal is strictly related to the accurate selection of the waveform shape by interferometers • the amplitudes of the strongest GW signals are expected or any possible detection tool. Such an achievement could to be roughly an order of magnitude smaller than LISA's give information on the nature of the GW source, on the instrumental noise. propagating medium, and, in principle, on the gravitational We investigate the GW emission by binary systems, in theory producing such a radiation [116]. the extreme mass ratio limit, by the quadrupole Considering the formulas of previous Section, the GW- approximation, considering orbits affected by both nutation amplitude can be evaluated by and precession effects, taking into account also gravitomagnetic terms in the weak field approximation of the 2G metric. We will see that gravitational waves are emitted with h jk (t, R)= Q

jk , (9.1) 4 a "peculiar" signature related to the orbital features: such a Rc signature may be a "burst" wave-form with a maximum in correspondence to the periastron distance or a modulated R being the distance between the source and the observer waveform, according to the orbit stability. Here we face this and, due to the above polarizations, { j, k} = 1,2 . The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 141

Fig. (18). Plots of z (t) (left upper panel) and z (t) (right upper panel). It is interesting to see the differences of about five orders of NO Grav magnitude between the two plots. At the beginning, the effect is very small but, orbit by orbit, it grows and, for a suitable interval of coordinated time, the effect cannot be neglected (see the left bottom panel in which the differences in x and y, starting from the initial orbits up to the last ones, by steps of about 1500 orbits, are reported). The internal red circle represents the beginning, the middle one is the intermediate situation (green) and the blue one is the final result of the correlation between
x versus
y , being
x = x  x and Grav NO
y = y  y . On the bottom right, it is shown the basic orbit. Grav NO

Fig. (19). Plot of the differences of total gravitational waveform h, with and without the gravitomagnetic orbital correction for a neutron star * of 1.4M orbiting around a MBH . The waveform has been computed at the Earth-distance from SgrA (the central Galactic Black Hole).
The example we are showing has been obtained solving the systems for the following parameters and initial conditions: μ
1.4M , r ,
0
1 1 E = 0.95 ,
=0,  = ,

=0,  =
r
and r
=
. It is worth noticing that frequency modulation gives cumulative effects after 0 0 0 0 0 0 2 10 100 suitable long times.

From Eq. (9.1), it is straightforward to show that, for a hxy = h yx =3 [cos 2 sin (4
cos sin )r 2 μ 


+ 
binary system where m
M and orbits have gravitomagnetic corrections, the Cartesian components of +2r
(2 cos 2 2
+

sin 2
sin 2)r GW-amplitude are sin hxx =2μ[(3 2 2
1)r
2 + 6r(
2 sin 2 cos sin cos 1 2 2 2 2 + sin 2(2r

sin  + r(2
cos 2
4
sin  2
 2 sin 2)r
+ r((3 2 2
1)r

sin cos sin 2 ( 2 2 +


sin 2
))r + r

sin 2], +3r(
cos 2 

sin 2 sin 2 sin cos where we are assuming geometrized units. The above
sin(sin( 2 cos 2 +  cos sin)


cos 2)))], cos formulas have been obtained from Eqs. (4.27), (4.28), (4.29). The gravitomagnetic corrections give rise to signatures on h yy =2μ[(3 2 2
1)r
2 + 6r( sin 2 2 sin sin sin the GW-amplitudes that, in the standard Newtonian orbital motion, are not present (see for example [32, 33]). On the +
sin 2 2)r
++r((3 2 2
1)r

sin sin sin other hand, as discussed in the Introduction, such corrections cannot be discarded in peculiar situations as dense stellar +3r(

2 cos 2
2 + 

sin 2
sin 2 sin clusters or in the vicinity of galaxy central regions. We are going to evaluate these quantities and results are shown in +sin
(


cos
2 + sin
( 2 cos 2 +  cos sin))))], Figs. (18, 19, 20, 21). sin 142 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis

Fig. (20). Plots along the panel lines from left to right of field velocities along the axes of maximum covariances, total gravitational emission waveform h and gravitational waveform polarizations h and h for a neutron star of 1.4M . The waveform has been computed for the +

Earth-distance from Sagittarius A (the central Galactic Black Hole SgrA * ). The plots we are showing have been obtained solving the system
1 1 for the following parameters and initial conditions: μ
1.4M , E = 0.95 ,
=0,  = ,

=0, 
=
r
and r
=
. From top
0 0 0 0 0 0 2 10 100 to bottom of the panels, the orbital radius is r =20μ,1500μ,2500μ . See also Table I. 0

Fig. (21). Plots along the panel lines from left to right of field velocities along the axes of maximum covariances, total gravitational emission waveform h and gravitational waveform polarizations h and h for a Black Hole (BH) of 10M . The waveform has been computed for the +

Earth-distance to SgrA * . The plots we are showing have been obtained solving the system for the following parameters and initial conditions:
1 1 μ
10M , E = 0.95 ,

=0 =0,  = ,

=0,  =
r
and r
=
. From top to bottom of the panels, the orbital radius is
0 0 0 0 0 0 0 2 10 100 r0 = 20μμμ ,1000 ,2500 . See also Table I. The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 143

Table 3. GW-amplitudes and Frequencies as Function of Eccentricity e , Reduced Mass μ , Orbital Radius r0 for the two Cases of Fiducial Stellar Objects m
1.4M and m
10M Orbiting Around a MBH of Mass M
3
106 M   

Fig. (22). Plot of estimated mean values of GW-emission in terms of strain h for two binary sources at the Galactic Center SgrA * with reduced mass μ
1.4M (red diamonds) and μ
10M (green circles). The blue line is the foreseen LISA sensitivity curve. The

* waveforms have been computed for the Earth-distance to SgrA . The examples we are showing have been obtained solving the systems for the parameters and initial conditions reported in Figs. 20, 21 and in Table I.

B. Numerical Results measured in the mass unit and scaling the distance according to the values shown in Table I. As it is possible to see in Now we have all the ingredients to estimate the effects of Table I, starting from r =20μ up to 2500μ , the orbital gravitomagnetic corrections on the GW-radiation. 0 Calculations have been performed in geometrized units in rmax
rmin order to evaluate better the relative corrections in absence of eccentricity e = evolves towards a circular orbit. r + r gravitomagnetism. For the numerical simulations, we have max min assumed the fiducial systems constituted by a m = 1.4M In Table I, the GW-frequencies, in mHz , as well as the h
amplitude strains and the two polarizations h and h are neutron star or m =10M massive stellar object orbiting +

shown. The values are the mean values of the GW h 6 * around a MBH M
3
10 M as SgrA . In the extreme h + h  amplitude strains ( h = max min ) and the maxima of the mass-ratio limit, this means that we can consider 2 mM polarization waves (see Figs. 20 and 21). In Fig. (22), the μ = of about μ
1.4M and μ
10M . The m + M

fiducial LISA sensitivity curve is shown [13] considering the computations have been performed starting with orbital radii confusion noise produced by White Dwarf binaries (blue 144 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis curve). We show also the h amplitudes (red diamond and number of encounters per unit time with impact parameter green circles for 1.4M and 10M respectively). It is less than b , which are suffered by a given star, is μ



3 worth noticing that, due to very high Signal to Noise Ratio, fd()vvaa times the volume of the annulus with radius b the binary systems which we are considering result and length v0 , that is, extremely interesting, in terms of probability detection, for the LISA interferometer (see Fig. 22). f (v )
b2v d 3v , (10.2)  a 0 a X. RATE AND EVENT NUMBER ESTIMATIONS IN where v0 = v
v a and v is the velocity of the considered DENSE STELLAR SYSTEMS star. The quantity in Eq. (10.2) is equal to 1/t for a star At this point, it is important to give some estimates of the coll number of events where gravitomagnetic effects could be a with velocity v : to obtain the mean value of 1/t , we coll signature for orbital motion and gravitational radiation. From average over v by multiplying (10.2) by f (v)/
, where the GW emission point of view, close orbital encounters, collisions and tidal interactions have to be dealt on average if  = f (v)d 3v is the number density of stars and the
we want to investigate the gravitational radiation in a dense 3 stellar system. On the other hand, dense stellar regions are integration is over d v . Thus the favored target for LISA interferometer [111] so it is 2 2 2 1  (v +v )/2 extremely useful to provide suitable numbers before its = e a  t 8 2 6  launching. coll
 To this end, it is worth giving the stellar encounter rate  4Gmr  producing GWs in astrophysical systems like dense globular r v
v + coll d 3vd 3v . (10.3) clusters or the Galactic Center. In general, stars are  coll a  a v
v a approximated as point masses. However, in dense regions of   stellar systems, a star can pass so close to another that they Replacing the variable v by V = v
v , the argument raise tidal forces which dissipate their relative orbital kinetic a a energy and the Newtonian mechanics or the weak field limit 2  1122 of GR cannot be adopted as good approximations. In some of the exponential is then

vV+ V /  , and if 24 cases, the loss of energy can be so large that stars form   binary (the situation which we have considered here) or 1 we replace the variable v by v = v
V (the center of multiple systems; in other cases, the stars collide and cm 2 coalesce into a single star; finally stars can exchange mass velocity), then one has gravitational interaction in non-returning encounters. 2 2 2 1  (v +V )/2  4Gmr  To investigate and parameterize all these effects, one has = e cm r V + coll dV . (10.4) 2 6  coll to compute the collision time t , where 1/t is the tcoll 8
  V coll coll collision rate, that is, the average number of physical The integral over v is given by collisions that a given star suffers per unit time. As a rough cm 2 2 approximation, one can restrict to stellar clusters in which all v / e cm d 3v =
3/2 3 . (10.5) stars have the same mass m .  cm Let us consider an encounter with initial relative velocity Thus v and impact parameter b . The angular momentum per 1/2 0 0 2 2 1   V /4 2 3 unit mass of the reduced particle is Lbv= . At the distance = e rcollV + 4GmVrcoll dV (10.6) 0 t 2 3 
() coll  of closest approach, which we denote by r , the radial coll velocity must be zero, and hence the angular momentum is The integrals can be easily calculated and then we find L = r v , where v is the relative speed at r . It is coll max max coll 1 4
Gmr easy to show that [25] =4
 r 2 + coll . (10.7) t coll  4Gmr coll b2 = r 2 + coll . (10.1) coll v2 The first term of this result can be derived from the 0 kinetic theory. The rate of interaction is 
V , where
is If we set r equal to the sum of the radii of the two coll the cross-section and V is the mean relative speed. stars, a collision will occur if the impact parameter is less than the value of b , as determined by Eq. (10.1). Substituting  =
r 2 and V =4 /
(which is coll The function f (v )d 3v gives the number of stars per appropriate for a Maxwellian distribution with dispersion a a
) we recover the first term of (10.7). The second term unit volume with velocities in the range v + d 3v . The a a represents the enhancement in the collision rate by The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 145 gravitational focusing, that is, the deflection of trajectories For a typical globular cluster around the Galactic Center, 9 1 by the gravitational attraction of the two stars. the expected event rate is of the order of 2 10
yrs
which may be increased at least by a factor 100 if one If r is the stellar radius, we may set r =2r . It is
* coll * considers the number of globular clusters in the whole convenient to introduce the escape speed from stellar Galaxy. If the stellar cluster at the Galactic Center is taken 2Gm into account and assuming the total mass M 3 106 M ,


surface, v* = , and to rewrite Eq. (10.7) as r
1 * the velocity dispersion

150 km s and the radius of the object R
10 pc (where
= 4.3), one expects to have 2 1  v 
5 = =16 r 2 1 * =16 r 2 (1 ), (10.8)
10 open orbit encounters per year. On the other hand, if 
 * + 2
 * +  t 4 6
1 coll   a cluster with total mass M
10 M , 
150 km s and
where R
0.1 pc is considered, an event rate number of the order of unity per year is obtained. These values could be v2 Gm realistically achieved by data coming from the forthcoming = * = (10.9)
2 2 space interferometer LISA. As a secondary effect, the above 4 2 r * wave-forms could constitute the "signature" to classify the is the Safronov number [25]. In evaluating the rate, we are different stellar encounters thanks to the differences of the considering only those encounters producing gravitational shapes (see Figs. 20 and 21). waves, for example, in the LISA range, i.e. between 10
4 and 10
1 Hz (see e.g. [117]). Numerically, we have 7. DISCUSSION, CONCLUSIONS AND PERSPECTIVES 3 10  v   10 pc  1 
5.510
yrs
We have considered the two-body problem in Newtonian 10kms
1 UA2  R  and relativistic theory of orbits in view of characterizing the gravitational radiation, starting from the motion of the
<< 1 (10.10) sources. We have reported several results concerning the 2 equations of motion, and the associated Lagrangian     
10 M v  formulation, of compact binary systems. These equations are 
5.510  105 M 10kms
1 UA2 necessary when constructing the theoretical templates for       searching and analyzing the GW signals from inspiralling 3 compact binaries in VIRGO-LIGO and LISA type 10 pc  1   yrs
 >> 1 (10.11) experiments. Considering the two-body problem, we mean R   the problem of the dynamics of two structureless, non- spinning point-particles, characterized by solely two mass If
>> 1 , the energy dissipated exceeds the relative parameters m and m , moving under their mutual, purely kinetic energy of the colliding stars, and the stars coalesce 1 2 into a single star. This new star may, in turn, collide and gravitational interaction. Surely this problem, because of its merge with other stars, thereby becoming very massive. As conceptual simplicity, is among the most interesting ones to its mass increases, the collision time is shorten and then be solved within any theory of gravity. Actually, there are there may be runaway coalescence leading to the formation two aspects of the problem: the first sub-problem consists of a few supermassive objects per clusters. If
<< 1 , much into obtaining the equation of the binary motion, the second of the mass in the colliding stars may be liberated and is to find the (hopefully exact) solution of that equation. We forming new stars or a single supermassive objects (see referred to the equation of motion as the explicit expression [118]). Both cases are interesting for LISA purposes. of the acceleration of each of the particles in terms of their Note that when one has the effects of quasi-collisions positions and velocities. It is well known that in Newtonian (where gravitomagnetic effects, in principle, cannot be gravity, the first of these sub-problems is trivial, as one can discarded) in an encounter of two stars in which the minimal easily write down the equation of motion for a system of N separation is several stellar radii, violent tides will raise on particles, while the second one is difficult, except in the two- the surface of each star. The energy that excites the tides body case N =2, which represents, in fact, the only comes from the relative kinetic energy of the stars. This situation amenable to an exact treatment of the solution. In effect is important for
>> 1 since the loss of small amount GR, even writing down the equations of motion in the of kinetic energy may leave the two stars with negative total simplest case N =2 is difficult. Unlike in Newton's theory, energy, that is, as a bounded binary system. Successive peri- it is impossible to express the acceleration by means of the center passages will dissipates more energy by GW positions and velocities, in a way which would be valid radiation, until the binary orbit is nearly circular with a within the exact theory. Therefore we are obliged to resort to negligible or null GW radiation emission. approximation methods. Let us feel reassured that plaguing Let us apply these considerations to the Galactic Center the exact theory of GR with approximation methods is not a which can be modelled as a system of several compact stellar shame. It is fair to say that many of the great successes of clusters, some of them similar to very compact globular this theory, when confronted to experiments and clusters with high emission in X-rays [119]. observations, have been obtained thanks to approximation 146 The Open Astronomy Journal, 2011, Volume 4 Mariafelicia De Laurentis methods. Furthermore, the beautiful internal wheels of GR a time reversal. By contrast, the even-orders, as 1PN, also show up when using approximation methods, which correspond to a dynamics which is conservative. often deserve some theoretical interest in their own, as they • GR is a non-linear theory (even in vacuum), and some require interesting mathematical techniques. Here we have part of the gravitational radiation which was emitted by the investigated the equation of the binary motion in the post- particles in the past scatters off the static gravitational field Newtonian approximation, i.e. as a formal expansion when generated by the rest-masses of the particles, or interacts the velocity of light c tends to infinity. As a consequence of gravitationally with itself. the equivalence principle, which is incorporated by hand in From all these considerations, the post-Newtonian Newton's theory and constitutes the fundamental basis of equations were also obtained, for the motion of the centers of GR, the acceleration of particle1 should not depend on m 1 mass of extended bodies, using a technique that can be (nor on its internal structure), in the test-mass limit where the qualified as more physical than the surface-integral method, mass m is much smaller than m . This is, of course, as it takes explicitly into account the structure of the bodies. 1 2 satisfied by the Newtonian acceleration, which is Particularly interesting is considering gravitomagnetic effects in the geodesic motion. In particular, one can independent of m1 , but this leaves the possibility that the consider the orbital effects of higher-order terms in v / c acceleration of the particle1, in higher approximations, does which is the main difference with respect to the standard depend on m , via the so-called self-forces, which vanish in approach to the gravitomagnetism. Such terms are often 1 discarded but, as we have shown, they could give rise to the test-mass limit. Indeed, this is what happens in the post- interesting phenomena in tight binding systems as binary Newtonian and gravitomagnetic corrections, which show systems of evolved objects (neutron stars or black holes). explicitly many terms proportional to (powers of) m1 . They could be important for objects falling toward extremely Though the approximations and corrections to the orbits are massive black holes as those seated in the galactic centers really a consequence of GR, they should be interpreted using [74, 75]. The leading parameter for such correction is the the common-sense language of Newton. That is, having ratio v / c which, in several physical cases cannot be simply chosen a convenient general-relativistic (Cartesian) discarded. For a detailed discussion see for example [120- coordinate system, like the harmonic coordinate system 123]. A part the standard periastron precession effects, such adopted above, we have express the results in terms of the terms induce nutations and are capable of affecting the coordinate positions, velocities and accelerations of the stability basin of the orbital phase space. As shown, the bodies. Then, the trajectories of the particles can be viewed global structure of such a basin is extremely sensitive to the as taking place in the absolute Euclidean space of Newton, initial angular velocities, the initial energy and mass conditions which can determine possible transitions to and their (coordinate) velocities as being defined with chaotic behaviors. Detailed studies on the transition to chaos respect to absolute time. Not only this interpretation is the could greatly aid in gravitational wave detections in order to most satisfactory one from a conceptual point of view, but it determine the shape, the spectrum and the intensity of the represents also the most convenient path for comparing the waves (for a discussion see [124, 125]). theoretical predictions and the observations. For instance, the Solar System dynamics at the first post-Newtonian level is In the second part of this review, we have summarized defined, following a recent resolution of the International many of the most important topics in the theory of GWs. Astronomical Union, in a harmonic coordinate system, the Linearized theory as described in is adequate to describe the Geocentric Reference System (GRS), with respect to which propagation of GWs and to model the interaction of GWs one considers the absolute motion of the planets and with our detectors. A variety of formalisms have been developed. satellites. But because the equations come from GR, they are endowed with the following properties, which make them •Newtonian theory The emission of gravitational waves truly relativistic. from stellar encounters in Newtonian regime interacting on elliptical, hyperbolic and parabolic orbits is studied in the • The one-body problem in GR corresponds to the quadrupole approximation. Analytical expressions are then Schwarzschild solution, so the equations possess the correct derived for the gravitational wave luminosity, the total perturbative limit, that given by the geodesics of the energy output and gravitational radiation amplitude produced Schwarzschild metric, when the mass of one of the bodies in tight impacts where two massive objects closely interact at tends to zero. an impact distance of 1AU . • Because GR admits the Poincaré group as a global •Post-Newtonian theory. PN theory is one of the most symmetry (in the case of asymptotically flat space-times), important of these formalisms, particularly for modeling the harmonic-coordinate equations of motion stay invariant binary systems. Roughly speaking, PN theory analyzes when we perform a global Lorentz transformation. sources using an iterated expansion in two variables: The • Since the particles emit gravitational radiation there are “gravitational potential”,

M / r , where M is a mass some terms in the equations which are associated with scale and r characterizes the distance from the source; and radiation reaction. These terms appear at the order 2.5 PN or velocities of internal motion, v . (In linearized theory, we
c 5 that we discarded in our discussion (where 5=2s +1 , assume
is small but place no constraints on v .) s =2 being the helicity of the graviton). They correspond to Newtonian gravity emerges as the first term in the an odd- order PN correction, which does not stay invariant in expansion, and higher order corrections are found as the The Newtonian and Relativistic Theory of Orbits The Open Astronomy Journal, 2011, Volume 4 147 expansion is iterated to ever higher order. Our derivation of The motion of particles in general relativity theory. Can J Math the quadrupole formula gives the leading order term in the 1949;1: 209. (c) Einstein A, Infeld L, Hoffman B. The Gravitational Equations and the Problem of Motion. Ann Math PN expansion of the emitted radiation. See [126] and 1938;39: 65-100. references therein for a comprehensive introduction to and [2] Droste J. The field of a single centre in EinsteinÕs theory of explication of this subject. gravitation, and the motion of a particle in that field. Versl K Akad Wet Amsterdam 1979;19: 447-55. •Gravitomagnetic corrections. The gravitomagnetic [3] de Sitter W. On Einstein's Theory of Gravitation and its effect could give rise to interesting phenomena in tight Astronomical Consequences.First Paper. Mon Not R Astr Soc 1916;76: 699-728. (a) de Sitter W. On EinsteinÕs Theory of binding systems such as binaries of evolved objects (NSsv or Gravitation and its Astronomical Consequences, Second Paper. BHs). The effects reveal particularly interesting if is in Mon Not R Astr Soc 1916;77 :155-184. the range (10

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Received: May 20, 2011 Revised: June 08, 2011 Accepted: June 10, 2011 © Mariafelicia De Laurentis; Licensee Bentham Open. This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.