Abstract:Binary pulsars are rapidly rotating highly magnetized neutron stars orbiting each other so fast and close together that they should, according to General Relativity, emit large amounts of gravitational radiation causing ripples in the space-time. An object disposed to gravitational radiation will become alternately longer and thinner, shorter and broader. The distance between objects will increase and decrease rythmically. The effects is, however, not very big, not even detected yet. Two neutron stars orbiting each other like this will lose energy due to the outgoing radiation and therefore the orbits will shrink and the periods shorten.The orbiting period of the binary pulsar PSR1913+16, discovered by Hulse and Taylor, has in fact been observed to decrease and the agreement with the prediction of General Relativity is better than 0,5%. This fact is considered as an indirect evidence for the gravitational radiation to exist.In this report, I will review some articles describing the post-Newtonian approximation for motion of binary pulsars. The articles are not containing any deeper derivations so neither will this paper. Contrary, I will just give the results obtained and refer to the source.

1 Introduction

1.1 Pulsars

1.2 Binary systems

1.3 Binary pulsars

1.4 Timing model

1.5 The motion of binary systems

2 Post-Newtonian Motion

2.1 The center of mass and the Lagrangian

2.2 Radial motion

2.3 Angular motion

2.4 Relative orbit

2.5 Motions of each body

3 Post-Newtonian Timing

3.1 Timing formula

3.2 Motion in terms of the proper time

4 Gravitational Waves

5 Summary

6 References

1 Introduction

1.1 Pulsars

Pulsars are rapidly rotating highly magnetized neutron stars and are formed by a Type II supernova explosion. A Type II supernova is characterized by having a spectrum with prominent hydrogen lines and is produced by a core collapse in a massive star whose outer layers were largely intact. A small, compact star may be left. Its gravity is then so strong that the electrons have been forced into the protons in the atomic nuclei and formed neutrons. The star has an enormous density but though only about 20 km in diameter, yet it weighs at least as much as our Sun. When charged particles are accelerated near a magnetized neutron star's magnetic poles, two oppositely directed beams of radiation are created (as shown in figure 1). The fastest pulsars spin several hundreds of revolutions per second and if the star's magnetic axis is tilted at an angle from the axis of rotation, the beam sweeps around the sky as the star rotates. If the Earth lies in the path of one of these beams, we detect radiation that appears to pulse on and off. The period of the pulses is therefore simply the rotation period of the neutron star.

1.2 Binary systems

About half of the visible stars in the night sky are not isolated individuals. Instead they are multiple-star systems, in which two or more stars orbit each other, called binary systems. These stars orbit each other because of the mutual gravitational attraction and, in the Newtonian thinking, their orbital motions obey Kepler's third law. If the orbit that one star appears to describe around the other is considered, Kepler's third law for binary star systems can be written as

(1.1)

where $M_{1}$ and $M_{2}$ are the masses of the stars,

is the semi-major axis of one star's orbit around the other and

is the orbital period.

1.3 Binary pulsars

About 4% of all known pulsars in the galactic disc are members of binary systems. Their orbiting companions are either white dwarfs, main sequence stars or other neutron stars. If two pulsars are massive and move around each other in tight orbits, they lose energy through a process called gravitational radiation. This is a prediction of Einstein's general theory of relativity and will briefly be discussed in section 4. As a result the two pulsars spiral toward each other and eventually collide, a process that may play an important role in seeding the interstellar medium with the building blocks of planets.

1.4 Timing model

Soon after their discovery it became clear that pulsars are excellent celestial clocks. The period can be measured to one part in $10^{13}$ over a few months leading to a source for applications used as time keepers and probes of relativistic gravity, like Gravity Probe B Note_1 .

In order to model the rotational motion of the neutron star, we need to measure the time of arrival in an inertial frame. An observatory on Earth experiences accelerations with respect to the neutron star due to the Earth's rotation and orbital motion around the Sun and is therefore not an inertial frame. The barycenter (center of gravity) of the solar system can, to a very good approximation, be regarded as an inertial frame. The transformation is summarized as the difference between the barycentric time of arrival $t_{a}$ and the observed time of arrival

and can be written as $\bigskip$

(1.2)

where $\vec{r}_{o}$ is the position of the observatory with respect to the barycenter, $\vec{s}$ is a unit vector in the direction of the pulsar at a distance

and

is the speed of light. The terms $\Delta _{E}$ and $\Delta _{S}$ represents the Einstein and Shapiro corrections due to general relativistic time delays in the solar system and will be discussed further in section 3.1. Measurements can be carried out at different observing frequencies with different dispersive delays (different frequencies propagate at different group velocities) so the times of arrival are generally referred to the equivalent time that would be observed at infinite frequency. This transformation is the term $\Delta _{D}$ .

1.5 The motion of binary systems

The ordinary non-relativistic two-body problem can be divided in two sub-problems. The first being the derivation of the orbital equations of motion for two gravitationally interacting bodies, and the second being the solution of these equations of motion. The first sub-problem can be simplified by approximating the orbital equations of motion of the bodies by the equations of motion of two point masses. By doing this, the second sub-problem can be exactly solved.

The two-body problem in General Relativity is not at all well posed and since the equations of motion are contained in the gravitational field equations it is not a simple thing to separate the problem in two sub-problems as in the non-relativistic case. Even if one could achieve such a separation and derive some equations of orbital motion for the two bodies, these equations would not be ordinary differential equations but some kind of retarded-integro-differential system Note_2 . This system can however be transformed into ordinary differential equations and, for widely separated, slowly moving, strongly self-gravitating bodies, expanded in power series of

. For the first post-Newtonian approximation, i.e. the first relativistic corrections to Newtons first law, the orbital equations of motion for these bodies depend only on two parameters having the dimensions of mass and are identical to the equations of motion for weakly self-gravitating bodies. At this point, the first sub-problem is managed and the second sub-problem is solved at the first post-Newtonian level.

2 Post-Newtonian Motion

The extremely precise tracking of the orbital motion of the Hulse-Talyor pulsar PSR1913+16 made it necessary to work out explicitly all the post-Newtonian effects in the motion. This section reviews the main parts of the method for solving this motion presented by Damour and Deruelle in [4].

2.1 The center of mass and the Lagrangian

The (first) post-Newtonian orbital equations of motion of a binary system can be derived from a Lagrangian with positions of the centers of mass $\vec{r}$ and $\vec{r}^{\prime }$ of the two bodies:

where

and $m^{\prime }$ are the mass parameters of the bodies,

is the velocity of light and

is Newton's constant. The notations

and

have been used. The total linear momentum of the system can be shown to be constant and is given by $\bigskip$

In a post-Newtonian center of mass frame where $\vec{P}_{PN}=0$ the center of mass expressions are given by

MATH

It can even be shown to be sufficient to use the non-relativistic center of mass expressions given by

where

, $M=m+m^{\prime }$ and

. The problem has now reduced to the much more simpler problem of solving the relative motion in the post-Newtonian center of mass frame. The resulting relative Lagrangian can then

be shown to be given by

with

where we have introduced

. The Lagrangian in equation (2.5) was obtained by Infeld and Plebanski in [3].

By using an approach closely following the standard methods for solving a non-relativistic two-body problem and by using polar coordinates,

and

, in the plane in which the motion takes place one can finally find the post-Newtonian equation of radial motion

MATH

and the post-Newtonian equation of angular motion

2.2 Radial motion

To solve the equation of radial motion we will now reduce the problem to the integration of an auxiliary, i.e. a "helping", non-relativistic radial motion. Using the change of variable

where

. Using the fact that

is of order $1/c^{2}$ and that we can neglect terms of order $1/c^{4}$ , replacing (2.11) in (2.6) gives us

.

The solution of (2.12) is retrieved by introducing a parametrization by means of the eccentric anomaly

. The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipse's circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse (figure 2).

In these formulas, $n=2\pi /period$ represents the mean motion and $a_{R}$ is the relative semi-major axis of the orbit. Furthermore, $e_{t}$ represents the time eccentricity and $e_{R}$ represents the relative radial eccentricity. The eccentricity may be interpreted as a measure of how much this shape deviates from a circle and the appearance of two different eccentricities is the main difference between the relativistic radial motion and the non-relativistic one. The relationship between these two is given by

By using (2.8), $a_{R}$ , $e_{R}$ , $e_{t}$ and

can be written in terms of

and

Here one sees that both the semi major axis and the mean motion depend only on the center of mass energy. This means that the well-known result of the Newtonian elliptic motion is still valid at the post-Newtonian level. In fact, the same is also true for the period of the orbit, $P=2\pi /n$ .

2.3 Angular motion

The problem of solving the equation of angular motion can, just as before, be reduced to the integration of an auxiliary non-relativistic angular motion. Let us make the following change of variable:

Comparing (2.16) and (2.21b) we see that $e_{t}$ and $\hat{e}$ differ only by a small term of order $1/c^{2}$ that can be neglected and so (2.23) is transformed to a Newtonian like equation of angular motion

Looking at (2.18) and (2.20) one can see that

has a minima (periastron passage) for

This, according to Damour and Deruelle, means that the periastron precesses by the angle

at each turn ([5]).

2.4 Relative orbit

This section will not contain any derivations but only the result of calculations made in [5]. By eliminating

between (2.14) and (2.25) one can show that the relative orbit is given by

2.5 Motions of each body

By using the solution for the relative motion,

and $\theta (u)$ in the post-Newtonian center off mass formulae we can obtain the expressions for the relativistic motions of each body. Like in the preceding section this one will not contain any derivations but just the resulting equations. The result is

3 Post-Newtonian Timing

This section reviews the results of the timing formula presented by Damour and Deruelle in [6].

3.1 Timing formula

The timing formula is a formula linking the time of arrival

of the Nth pulse emitted by a pulsar in a binary system to the integer N. Introducing the time variables $\tau _{a}$ , $t_{a}$ , $t_{e}$ , $T_{e}$ and

will allow a derivation of a timing formula of the type

Assuming that using

one has computed the time of arrival $\tau _{a}$ of the N $^{th}$ pulse at the barycenter of the solar system in absence of any solar gravitational redshift and interstellar dispersion Note_4 . We can hereby define $\tau _{a}\equiv t$ to be the infinite-frequency barycenter arrival time.

Using a system of coordinates such that the barycenter of the binary system is at rest at the origin we can compute the coordinate time of arrival $t_{a}$ , a relation linking the proper time $\tau _{a}$ to the integer N. Here the barycenter of the solar system is moving with the velocity $\vec{v}_{b}$ and the relation (3.1b) is given by

In this context, constants like the one above are unimportant and will be neglected.

By using a coordinate position vector of the barycenter of the solar system $\vec{r}_{b}$ and a coordinate position vector of the pulsar $\vec{r}$ (see figure 3), the coordinate time of arrival $t_{a}$ can be linked to the coordinate time of emission of the pulse $t_{e}$ by

Now, to find the relation (3.1d), we first introduce a suitable proper time

for the pulsar and a proper time $T_{e}$ for the emission of the Nth pulse. These two proper times are related as

where $\Delta _{A}$ is the aberration time delay, a time delay associated to the angular shift of the proper angle measuring the position of the emission spot which rotates around the spin axis. This equation gives the time at which the Nth pulse would have been emitted if the pulsar mechanism had been a radial pulsation instead of a rotating beacon. From this fact, one finds that

is implicitly defined as a function of

by the relation

where $\nu$ is the proper rotation frequency of the pulsar (at

). The coordinate time of emission, $t_{e}$ , is now linked to the proper time of emission $T_{e}$ by

where $\Delta _{E}$ is the Einstein time delay, a delay caused by the gravitational redshift due to the companion and by the second order Doppler effect.

where $\Delta _{R}$ is the Roemer time delay, i.e. the time of flight across the orbit counted from the barycenter and projected on the line of sight. In this formula, the order of magnitude of the four different time delays are

3.2 Motion in terms of the proper time

We are using a center of mass frame such that the motion of the pulsar lies in the plane

. By choosing the plane of the sky to be

the orientation of the center of mass frame with respect to this plane of the sky are given by the two angles

and

where $\Omega$ is the longitude of the ascending node Note_6 and

is the inclination. The triad

can be found from the reference triad

by two successive rotations (illustrated in figure 4). First one rotates the triad

into a temporary triad

one ends up at the triad

. In this sense, the unit vector $\vec{e}_{x}$ is directed towards the ascending node

and

is pointing from the Earth to the orbit of the pulsar.

By using planar polar coordinates and looking back at equations (2.13), (2.14) and (2.25) one can find a parametric representation of the motion of the pulsar given by

where

are constants. Using equation (3.6) with (3.11) leads to a relation between the eccentric anomaly parameter

and the proper time

The parametric representation of the motion of the pulsar expressed in proper time

is now given by the equations (3.14), (3.12) and (3.13), so what is left now is to find an explicit timing formula. This turns out to be quite complicated and will thus not be presented in this project. The interested reader is recommended to have a look at [6].

4 Gravitational Waves

The theory of General Relativity, postulated by Einstein, describes the gravitational force as a consequence of the curvature of space-time. This curvature is caused by massive objects; the more massive an object is, the greater curvature it causes and the more intense the gravity is. A massive object, like a member of a binary system, rotating around its companion will cause ripples in space-time, just like the ripples in a lake. These ripples are referred to as gravitational waves. As these waves evolves through the universe, space-time will distort in a peculiar way. The distance between objects will increase and decrease rhythmically as the wave passes by. This effect is not very big though, not even detected yet.

One important source of radiation is the Hulse-Taylor binary PSR1913+16, a binary system of two stars, one of which is a pulsar and the other probably an ordinary neutron star. Here one pulsar year is only about eight hours and, by observing the shift in the pulses, the stars are found to be equally heavy, each weighing about 1,4 times as much as the Sun. As the energy is carried away from the binary by the gravitational wave, the orbits will converge and with time the members will collide. This kind of orbit is called an inspiral and can be observed by the pulsar timing of the system. These observations are the first indirect evidence for gravitational waves but a more interesting observation would be a direct evidence. This would provide us with a rigorous test of the theory of General Relativity and also information about things we cannot see with electromagnetic radiation, like black holes. Nevertheless, the weak nature of gravitational radiation makes it very difficult to design a sensitive detector filtering out the noisy background. Various detectors are though in use and we expect a result to appear in a near future. There is a lot more to learn about gravitational waves and for the reader interested in an introduction to the topic I recommend to have a look at [4].

5 Summary

In section 2, we reviewed the main parts of the method for solving the post-Newtonian motion in the post-Newtonian mass frame. We found the parametric solution to be

In section 3, we reviewed the motion of the pulsar expressed in proper time

which are given in a parametric form by

6 References

[1] Freedman and Kaufmann, Universe, 7 $^{th}$ edition, New York, W. H. Freeman and Company, 2005

[2] Goldstein, Poole and Safko, Classical Mechanics, 3 $^{rd}$ edition, San Fransisco, Pearson Education, 2002

Contents

1 Introduction

1.1 Pulsars

1.2 Binary systems

1.3 Binary pulsars

1.4 Timing model

1.5 The motion of binary systems

2 Post-Newtonian Motion

2.1 The center of mass and the Lagrangian

2.2 Radial motion

2.3 Angular motion

2.4 Relative orbit

2.5 Motions of each body

3 Post-Newtonian Timing

3.1 Timing formula

3.2 Motion in terms of the proper time

4 Gravitational Waves

5 Summary

6 References