Binary Pulsars. By: Kristo er Hultgren. Karlstad University Faculty of Technology and Science Department of Physics

Transcription

1 Binary Pulsars By: Kristo er Hultgren Karlstad University Faculty of Technology and Science Department of Physics Course: Analytical Mechanics (FYSCD2) Examiner: Prof. Jürgen Fuchs January 5, 2007 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 e ects 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 PSR93+6, 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.

2 Contents Introduction 3. Pulsars Binary systems Binary pulsars Timing model The motion of binary systems Post-Newtonian Motion 6 2. The center of mass and the Lagrangian Radial motion Angular motion Relative orbit Motions of each body Post-Newtonian Timing 3 3. Timing formula Motion in terms of the proper time Gravitational Waves 8 5 Summary 9 6 References 20 2

3 Introduction. 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 gure ). 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 o. The period of the pulses is therefore simply the rotation period of the neutron star. Figure : A rotating, magnetized neutron star. 3

4 .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 M + M 2 = a3 P 2 (.) where M and M 2 are the masses of the stars, a is the semi-major axis of one star s orbit around the other and P is the orbital period..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 brie y 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..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 0 3 over a few months leading to a source for applications used as time keepers and probes of relativistic gravity, like Gravity Probe B. 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 Gravity Probe B is the relativity gyroscope experiment being developed by NASA and Stanford University to test predictions of Albert Einstein s general theory of relativity. The experiment will check, very precisely, tiny changes in the direction of spin of four gyroscopes contained in an Earth satellite orbiting at 400-mile altitude directly over the poles. So free are the gyroscopes from disturbance that they will provide an almost perfect space-time reference system. They will measure how space and time are warped by the presence of the Earth, and, more profoundly, how the Earth s rotation drags space-time around with it. ([0]) 4

5 summarized as the di erence between the barycentric time of arrival t a and the observed time of arrival t and can be written as t a t = ~r o ~s c + (~r ~s)2 j~rj 2 2cd + E + S D (.2) where ~r o is the position of the observatory with respect to the barycenter, ~s is a unit vector in the direction of the pulsar at a distance d and c is the speed of light. The terms E and 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.. Measurements can be carried out at di erent observing frequencies with di erent dispersive delays (di erent frequencies propagate at di erent group velocities) so the times of arrival are generally referred to the equivalent time that would be observed at in nite frequency. This transformation is the term D..5 The motion of binary systems The ordinary non-relativistic two-body problem can be divided in two subproblems. The rst 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 rst sub-problem can be simpli ed 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 eld 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 di erential equations but some kind of retarded-integro-di erential system 2. This system can however be transformed into ordinary di erential equations and, for widely separated, slowly moving, strongly self-gravitating bodies, expanded in power series of v=c. For the rst post-newtonian approximation, i.e. the rst relativistic corrections to Newtons rst 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 rst sub-problem is managed and the second sub-problem is solved at the rst post-newtonian level. 2 An integro-di erential equation is an integral equation in which various derivatives of the unknown function can also be present. 5

6 2 Post-Newtonian Motion The extremely precise tracking of the orbital motion of the Hulse-Talyor pulsar PSR93+6 made it necessary to work out explicitly all the post-newtonian e ects in the motion. This section reviews the main parts of the method for solving this motion presented by Damour and Deruelle in [4]. 2. The center of mass and the Lagrangian The ( rst) post-newtonian orbital equations of motion of a binary system can be derived from a Lagrangian with positions of the centers of mass ~r and ~r 0 of the two bodies: L P N (~r(t); ~r 0 (t); ~v(t); ~v 0 (t)) = L N + c 2 L 2 (2:a) with L N = 2 mv2 + 2 m0 v 02 + Gmm0 R (2:b) and L 2 = 8 mv4 + 8 m0 v 04 + Gmm0 R 3v 2 + 3v 02 7(vv 0 ) (Nv)(Nv 0 ) G m + m0 R (2:c) where m and m 0 are the mass parameters of the bodies, ~v = d~r dt, R ~ = ~r ~r 0, R ~N = ~, c is the velocity of light and G is Newton s constant. The notations R ~v ~v = j~vj 2 = v 2 and ~v ~v 0 = (vv 0 ) have been used. The total linear momentum of the system can be shown to be constant and is given by ~P P N P P 0 (2:2) In a post-newtonian center of mass frame where ~ P P N = 0 the center of mass expressions are given by ~r = m ~ R + (m m0 ) 2M 2 c 2 ~r 0 = m 0 ~ R + (m m0 ) 2M 2 c 2 V 2 V 2 GM ~R (2:3a) R GM ~R (2:3b) R It can even be shown to be su cient to use the non-relativistic center of mass expressions given by 6

7 ~r = m ~ R (2:4a) ~r 0 = m 0 ~ R (2:4b) ~v = m ~ V (2:4c) ~v 0 = m 0 ~ V (2:4d) where V ~ = d R=dt ~ = ~v ~v 0, M = m + m 0 and = mm 0 =M. 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 L R P N ( ~ R; ~ V ) = 2 V 2 + GM R + c 2 L 2 ~R; ~ V m ; V ~! m 0 (2:5a) with L 2 = 4 ( 3)V 8 c 2 + GM 2Rc 2 (3 + )V 2 + (NV ) 2 GM R (2:5b) where we have introduced = =M = mm 0 =(m + m 0 ) 2. 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, R x = R cos and R y = R sin, in the plane in which the motion takes place one can nally nd the post-newtonian equation of radial motion 2 dr = A + 2B dt R + C R 2 + D R 3 (2:6) and the post-newtonian equation of angular motion d dt = H R 2 + I R 3 (2:7) where the coe cients A, B, C, D, H and I are the following polynomials: A = 2E + 32 (3 ) Ec 2 (2:8a) B = GM + (7 6) Ec 2 (2:8b) 7

8 C = J 2 + 2(3 ) E c 2 + (5 0) G2 M 2 c 2 (2:8c) D = ( 3 + 8) GMJ 2 c 2 H = J + (3 ) Ec 2 (2:8d) (2:8e) I = (2 4) GMJ c 2 (2:8f) Here, the center of mass energy E and angular momentum J = j ~ Jj is given by E = 2 V 2 4 GM R +3 ( 3)V 8 c 2 + GM 2Rc 2 (3 + )V 2 + (NV ) 2 + GM R ~J = R ~ V ~ ( 3)V c 2 + (3 + )GM Rc 2 (2:0) (2:9) 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 R = ~ R + D 2C 0 (2:) where C 0 = lim c! C = J 2. Using the fact that D is of order =c 2 and that we can neglect terms of order =c 4, replacing (2.) in (2.6) gives us where d R ~! 2 = A + 2B dt ~R + C ~ (2:2a) ~R 2 ~C = C BD C 0 (2:2b). The solution of (2.2) is retrieved by introducing a parametrization by means of the eccentric anomaly u. 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 ( gure 2). The solution is given by 8

9 Figure 2: Eccentric anomaly where n(t t 0 ) = u e t sin u (2:3) R = a R ( e R cos u) (2:4) n = ( e t = A)3=2 B A B 2 C (2:5a) BD =2 C 0 (2:5b) a R = e R = B A + AD 2C 0 (2:5c) + AD e t 2BC 0 (2:5d) In these formulas, n = 2=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 9

10 be interpreted as a measure of how much this shape deviates from a circle and the appearance of two di erent eccentricities is the main di erence between the relativistic radial motion and the non-relativistic one. The relationship between these two is given by e R e t = + GM a R c = + (3 8) E c 2 (2:6) By using (2.8), a R, e R, e t and n can be written in terms of E and J: a R = e R = e t = n = ( GM 2E 2 ( 7) E c 2 (2:7a) + 2E 5 G 2 M E J 2 2 c 2 + ( 6) G2 M 2 c 2 + 2E 7 G 2 M E)3=2 GM 4 ( 5) E c 2 E c 2 =2 J 2 + ( 2 + 2) G2 M 2 (2:7d) c 2 =2 (2:7b) (2:7c) 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=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: R = ^R + I 2H (2:8) Replacing (2.8) in (2.7) gives us then d dt = H^R2 (2:9) where we can use ^R as ^R = ^a( ^e cos u) (2:20) 0

11 with ^a = a R ^e = e R I 2H AI 2BH (2:2a) Using (2.3) we see that (2:2b) dt = n ( e t cos u)du (2:22) and hence d = H e t cos u n^a 2 du (2:23) ( ^e cos u) 2 Comparing (2.6) and (2.2b) we see that e t and ^e di er only by a small term of order =c 2 that can be neglected and so (2.23) is transformed to a Newtonian like equation of angular motion d = H du n^a 2 ^e cos u (2:24) which can be integrated to 0 = KA^e (u) (2:25) where A^e (u) = 2 arctan " # =2 + ^e tan u ^e 2 (2:26) K = H= n^a 2 ( ^e 2 ) =2 (2:27). Looking at (2.8) and (2.20) one can see that R has a minima (periastron passage) for u = 0; 2; 4; ::: This, according to Damour and Deruelle, means that the periastron precesses by the angle = 2(K ) at each turn ([5]).

12 2.4 Relative orbit This section will not contain any derivations but only the result of calculations made in [5]. By eliminating u between (2.4) and (2.25) one can show that the relative orbit is given by with R = a R( e 2 R ) + e R cos f 0 (2:28a) f 0 = f + 2(e 2f =e R ) sin f (2:28b) where f = 0 K (2:28c) e 2f = G 4 e2 R a R ( e 2 (2:28d) R )c2 2.5 Motions of each body By using the solution for the relative motion, t(u), R(u) and (u) in the post- Newtonian center o 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 where r = ~a r ( e r cos u) (2:29a) a r = m0 M a R (2:29b) The orbit in space of one of the bodies can be described by Gm r = a 2 m 0 ^e r 2 2M 2 c 2 + ^e cos 0 K + Gm2 m 0 2M 2 c 2 This equation describes the conchoid 3 of a precessing ellipse. (2:29a) 3 A conchoid is a curve derived from a xed point O, another curve, and a length d. For every line through O that intersects the given curve at A the two points on the line which are d from A are on the conchoid. ([8]) 2

13 3 Post-Newtonian Timing This section reviews the results of the timing formula presented by Damour and Deruelle in [6]. 3. Timing formula The timing formula is a formula linking the time of arrival (Earth) a of the Nth pulse emitted by a pulsar in a binary system to the integer N. Introducing the time variables a, t a, t e, T e and T will allow a derivation of a timing formula of the type (Earth) a = f ( a ) (3:a) a = f 2 (t a ) t a = f 3 (t e ) t e = f 4 (T e ) T e = f 5 (T ) T = f 6 (N): (3:b) (3:c) (3:d) (3:e) (3:f) Assuming that using (Earth) a one has computed the time of arrival a of the N th pulse at the barycenter of the solar system in absence of any solar gravitational redshift and interstellar dispersion 4. We can hereby de ne a t to be the in nite-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 a to the integer N. Here the barycenter of the solar system is moving with the velocity ~v b and the relation (3.b) is given by a = f 2 (t a ) = ( ~v 2 b =c2 ) =2 t a + const: (3:2) 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 ~r b and a coordinate position vector of the pulsar ~r (see gure 3), the coordinate time of arrival t a can be linked to the coordinate time of emission of the pulse t e by 4 Dispersion occurs because the group velocity of the pulsed radiation through the ionised component of the interstellar medium is frequency dependent. Pulses emitted at lower radio frequencies travel slower through the interstellar medium, arriving later than those emitted at higher frequencies. 3

14 t a = f 3 (t e ) = t e + c j~r b(t a ) ~r(t e )j + S (3:3) where S is the Shapiro time delay 5. Figure 3: The coordinates of the barycenters. Now, to nd the relation (3.d), we rst introduce a suitable proper time T for the pulsar and a proper time T e for the emission of the Nth pulse. These two proper times are related as T := T e A (3:4) where 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 nds that T is implicitly de ned as a function of N by the relation 5 The Shapiro time delay is one of the four classic solar system tests of General Relativity. Electromagnetic signals passing near a massive object takes, due to the curved space-time, a slightly longer time to travel to a target than it would if the mass of the object were not present. 4

15 N = N 0 + T + 2 _T T 3 (3:5) where is the proper rotation frequency of the pulsar (at T = 0). The coordinate time of emission, t e, is now linked to the proper time of emission T e by t e = T e + E (3:6) where E is the Einstein time delay, a delay caused by the gravitational redshift due to the companion and by the second order Doppler e ect. By using equations (3.4), (3.6) and a Doppler factor D := p ~n ~v b=c (3:7) ~v 2 b =c 2 it can be shown that a is related to T by D a = T + R + E + S + A (3:8a) where R is the Roemer time delay, i.e. the time of ight across the orbit counted from the barycenter and projected on the line of sight. In this formula, the order of magnitude of the four di erent time delays are R c (3:8b) E c 2 (3:8c) S c 3 (3:8d) A c 3 (3:8e) 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 (~e x ; ~e y ). By choosing the plane of the sky to be (~e X0 ; ~e Y0 ) the orientation of the center of mass frame with respect to this plane of the sky are given by the two angles 2 [0; 2) and i 2 [0; ) where is the longitude of the ascending node 6 and i is the inclination. The triad (~e x ; ~e y ; ~e z ) can be found from the reference triad (~e X0 ; ~e Y0 ; ~e Z0 ) by two successive rotations (illustrated 6 The intersection between a plane in space and an orbital plane are called the line of nodes, where a node is the point where the orbit intersects the plane in space. An ascending node is then the node where the object enters from below into the upper hemisphere. ([2]) 5

16 in gure 4). First one rotates the triad (~e X0 ; ~e Y0 ; ~e Z0 ) into a temporary triad (~e X ; ~e Y ; ~e Z ) by ~e X = cos ~e X0 + sin ~e Y0 ; (3:9a) ~e Y = sin ~e X0 + cos ~e Y0 ; (3:9b) ~e Z = ~e Z0 : (3:9c) Then, by rotating this temporary triad as ~e x = ~e X ; (3:0a) ~e y = cos i~e Y + sin i~e Z ; (3:0b) ~e z = sin i~e Y + cos i~e Z ; (3:0c) one ends up at the triad (~e x ; ~e y ; ~e z ). In this sense, the unit vector ~e x is directed towards the ascending node P and ~e Z0 = ~n is pointing from the Earth to the orbit of the pulsar. Figure 4: Angular elements of the pulsar orbit 6

17 By using planar polar coordinates and looking back at equations (2.3), (2.4) and (2.25) one can nd a parametric representation of the motion of the pulsar given by n(t T 0 ) = u e t sin u (3:) r = a r ( e r cos u) (3:2) =! 0 + ( + k)a^e (u) (3:3) where n; T 0 ; e t ; a r ; e r ;! 0 ; k; ^e are constants. Using equation (3.6) with (3.) leads to a relation between the eccentric anomaly parameter u and the proper time T : n(t T 0 ) = u e T sin u (3:4) where e T is a modi ed expression given by e T = e t ( + ) (3:5a) with = G m 0 (m + 2m 0 ) c 2 a R M (3:5b) Here a R is given by equation (2.29b). The parametric representation of the motion of the pulsar expressed in proper time T is now given by the equations (3.4), (3.2) and (3.3), so what is left now is to nd 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]. 7

18 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 e ect is not very big though, not even detected yet. Figure 5: Gravitational Waves One important source of radiation is the Hulse-Taylor binary PSR93+6, 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,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 rst 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 di cult to design a sensitive detector ltering 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]. 8

19 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 n(t t 0 ) = u e t sin u (5:a) R = a R ( e R cos u) (5:b) r = a r ( e r cos u) (5:c) r 0 = a r 0( e r 0 cos u) (5:d) 0 = 2K arctan " # =2 + ^e tan u ^e 2 (5:e) where a R = n = ( GM 2E 2E)3=2 GM 2 ( 7) E c 2 4 ( 5) E c 2 (5:2a) (5:2b) and K; ^e; e t; e R ; e r ; a r ; e r 0; a r 0 are given in terms of the total energy and the total angular momentum. In section 3, we reviewed the motion of the pulsar expressed in proper time T which are given in a parametric form by n(t T 0 ) = u e T sin u (5:3a) r = a r ( e r cos u) (5:3b) =! 0 + 2( + k) arctan " # =2 + ^e tan u ^e 2 (5:3c) where e T = e t ( + ) (5:4a) and! 0 ; k are constants. 9

20 6 References [] 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 [3] Infeld and Plebanski, Motion and Relativity, Pergamon, Oxford, 960 [4] Chakrabarty, Gravitational Waves: An Introduction, Retrieved December, 2006, from [5] Damour and Deruelle, General relativistic celestial mechanics of binary systems. I. The post-newtonian motion, Retrieved December, 2006, from [6] Damour and Deruelle, General relativistic celestial mechanics of binary systems. II. The post-newtonian timing formula, Retrieved December, 2006, from [7] Lorimer, Binary and Millisecond Pulsars, Retrieved December, 2006, from [8] Conchoid (mathematics), Retrieved December, 2006, from [9] Eccentric anomaly, Retrieved December, 2006, from [0] Gravity Probe B, Retrieved December, 2006, from [] Gravitational wave, Retrieved December, 2006, from [2] The Nobel Prize in Physics 993, Retrieved December, 2006, from All images have been produced by the author. 20