Magnetic field decay in neutron stars: Analysis of general relativistic effects

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 61, Issue 12, 2000, Pages

  Zannias, Thomas ^c  

^a ASTROPHYSIKALISCHES INSTITUT POTSDAM   (Germany)

^b UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO   (Mexico)

^c UNIVERSIDAD MICHOACANA DE SAN NICOLÁS DE HIDALGO   (Mexico)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17144425040     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.61.123004     Document Type: Article

References (52)

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3. See, for instance, discussion in Ya. Zeldovich, A. A. Ruzmaikin, and D. D. Socolov, Magnetic Fields in Astrophysics (Gordon and Breach, New York, 1983).
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5. A recent discussion concerning possible mechanism leading to the creation of Pulsar’s Magnetic field can be found in
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9. For an introduction, see, A. G. Lyne and F. Graham-Smith, Pulsar Astronomy (Cambridge University Press, Cambridge, England, 1990).
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11. S. SenguptaAstrophys. J. 501, 792 (1998).
12. For a discussion of the dipole magnetic field on a Schwarzschild background see, for instance
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14. J. L. Anderson and J. M. Cohen, Astrophys. Space Sci. 9, 146 (1970)
15. I. Wasserman and S. L. Shapiro, Astrophys. J. 265, 1036 (1983).
16. There are some sensitive issues involved in the use of the terms “influence of general relativistic effects” and “B field decay.” It would be perhaps more correct to write that we are interested in studying how certain solutions of Maxwell’s equations expressed relative to global Lorentz frames and describing a particular physical situation would appear once they have the same physical problem considered in the presence of nonvanishing spacetime curvature. As far as the term “influence of general relativistic effects” is concerned here, rather than at the start, we are interested in considering the effects of spacetime curvature. Maxwell’s equations on flat spacetime can be written in an arbitrary curvilinear system and the issue how a B-decaying field relative to inertial frames would be perceived in noninertial observers, naturally arises. The present paper is not intended to analyze such an issue. Toward the end of the paper we shall make a few relevant comments pertinent to that issue.
17. R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
18. C. Misner, K. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).
19. A note on our conventions: Maxwell’s equations [apart from the insertion of the c factor in Eq. (2.1a)], signature of metric, curvature, etc., are the same as that of Ref. 12. In addition in the present work all Greek indices are assumed to be four dimensional, while Latin indices are spatials. The definition and normalization of the four- and three-dimensional totally antisymmetric Levi-Civita tensor density is the same as that of Ref. 12.
20. H. Weyl, Space-Time-Matter (Dover, New York, 1952).
21. We shall ignore in the present work possible additional effects of the high B field leading into a conductivity tensor. For simplicity we shall work with a scalar conductivity, leaving the complete treatment to future work.
22. Such a form of Maxwell’s electrodynamics corresponds to a 3+1 formulation. Early work on such approach can be found in
23. A. L. Zel’manov, Dokl. Akad. Nauk SSSR 107, 805 (1956)
24. F. B. Estabrook and H. D. Wahlquist, J. Math. Phys. 5, 1679 (1964)
25. G. F. R. Ellis, in Gargese Lectures in Physics, edited by E. Schatzman (Gordon and Breach, London, 1973), Vol. 6
26. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, New York, 1975).A historical account of those attempts can be found in Ref. 18 below.
27. In addition to the above references on (Formula presented) splitting of curved spacetime electrodynamics, the following reference discusses at great length and generality such a splitting and provides a flexible framework tailored towards astrophysical applications. Furthermore, in that reference the “absolute space” formalism has been first introduced
28. K. S. Thorne and D. Macdonald, Mon. Not. R. Astron. Soc. 198, 339 (1982).In fact we could deduce formulas (2.5) used in the main text by directly appealing to the equations of this work, appropriately restricted to our problem. However, for completeness purposes only, we briefly perform a (Formula presented) splitting of Maxwell’s equation tailored to the static background geometries.
29. The MHD approximation employed in the present paper is well suited for astrophysical plasmas and particularly for neutron star matter where typically (Formula presented). To see that let us suppose that (Formula presented) is a solution of Eqs. (2.8a2.8b2.8c) subject to the condition (Formula presented). If by R we denote the typical length scale of the system, then Eq. (2.8c) implies that in order of magnitude, (Formula presented), where T is the dynamical evolution time scale for the (Formula presented) fields and in arriving at that estimate we have taken Z to be of order unity. On the other hand (Formula presented) and thus taking into account Ohm’s law in Eq. (2.8b) one obtains (Formula presented). It follows then from this estimate, if the evolution time scale T is of the order of the Ohmic time, i.e., (Formula presented) then for the typical neutron star parameters (Formula presented) and for such situations the displacement current can be neglected from the right-hand side of Eq. (2.8b) implying that the MHD approximation is well justified. Since essentially in the MHD approximation one neglects the electromagnetic radiation, that point will allow us further on to join the interior solutions with static exterior dipole field.
30. Black holes: The Membrane Paradigm, edited by K. Thorne, R. H. Price and D. A. Macdonald (Yale University Press, New Haven, 1986). In that collection of articles the original “absolute space” approach of curved electrodynamics of Ref. 18 has been elaborated further and extensively applied to black-hole spacetimes. We have been charmed by the practical usefulness of this approach and in that spirit we have written Maxwell’s equations in the form (2.8). For the present case the “absolute space” is identified by the three-dimensional spacelike sections perpendicular to the Killing field. We should stress, however, that there are many more advantages of the “absolute space” formulation of curved spacetime electrodynamics than its mere practical usefulness, and the interested reader is referred to the above volume for more detailed applications.
31. Actually even for a time-dependent (Formula presented), the toroidal and poloidal component evolve independently of each other. In the present work we shall use in Sec. III a uniform conductivity, however, as we go along we shall point out implications on the evolution of the magnetic field components due to a conductivity characterized by an arbitrary spacetime dependence.
32. In the case of an arbitrary one-pole field one recovers an identical equation as the above. The sole exception is that in the factor of 2, in last term of the right-hand side is replaced by (Formula presented), respectively.
33. See, for instance, Y. Sang and G. Chanmugam, Astrophys. J., Lett. Ed. 323, L61 (1987).
34. As far as we are aware the so-called “Stoke’s function” has been introduced as a convenient parametrization of the poloidal fields in P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
35. Strictly speaking, the interior field for both the flat and curved space case, should be joined with a radiating into empty space solution of Maxwell’s equations. However, due to the MHD approximations and due to the long times evolved in the decay of the interior field, typically, and to a good approximation, one considers the exterior dipole field to consist of a sequence of quasistatic dipole solutions. Thus, essentially, we take the exterior dipole magnetic moment to be a slow varying function of time and that approximation allows us to perform the (Formula presented) mashing of the B across the star’s surface.
36. Exact solutions of this equation have been obtained, for example, in Ref. 23 above. The sequence of eigenvalues have been found long ago, see, for instance, H. Lamb, Philos. Trans. R. Soc. London 174, 519 (1883).
37. Early attempts to take into account the influence of the electromagnetic stresses on the structure of neutron stars can be found in G. Dautcourt and K. Fritze, Astron. Nachr. 295H, 211 (1971). More recently a large scale numerical computation of the structure of rotating neutron stars taking into account the effects of the Maxwell field has been performed by
38. M. Bocquet, S. Bonazolla, E. Gourgoulhon, and J. Novak, Astron. Astrophys. 301, 757 (1995). According to the results of this study and under the assumption that the B field is purely poloidal, the effect of the Maxwell field does not yield appreciably different neutron stars models than the conventional models for field strengths (Formula presented). For larger fields they report differences than the conventional models but issues related to the stability of such models have not been addressed yet.
39. D. Page, U. Geppert, and T. Zannias (in preparation).
40. W. H. Press, B. P. Flanery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University of Press, Cambridge, England, 1986);also Web site: http://www.nr.com
41. It should be stressed, however, that knowledge of all eigenvalues yields information regarding the behavior of the magnetic field before the exponentially decreasing phase is reached. Accordingly, construction of the other eigenvalues is a worthwhile project.
42. M. Prakash, Neutron Stars, in Nuclear and Particle Astrophysics, edited by J. G. Hirsch and D. Page (Cambridge University Press, Cambridge, England, 1998).
43. It should be stressed here that even though we are drawing physical conclusions based on the magnetic field measured by Killing observers, the influence of the curvature can be seen and described in a coordinate and observer free manner. One, for instance, may consider the field invariant (Formula presented) and examine its properties on a flat and curved spacetime. Within the MHD approximation, (Formula presented) field can be computed via Ohm’s law and thus the right-hand side of this field invariant is expressible in terms of the corresponding Stokes functions. In fact, for our case since (Formula presented) and current (Formula presented), are purely toroidal, they vanish on the poles of the star, hence (Formula presented). Thus Fig. 11 also supplies information regarding the behavior of this field invariant for the curved-flat space, respectively. Accordingly the “flat value” of (Formula presented) decays more rapidly than its curved counterpart. We have chosen to indicate the effect in terms of (Formula presented) since the latter is directly related to terrestrial observations.
44. In our numerical integration of Eq. (3.13a) for the cases (Formula presented) and (Formula presented), respectively, we have used as the boundary condition the expression (3.18). Although this is not entirely correct, numerically we have found that the content of Fig. 44 is rather insensitive to small changes in the surface boundary condition.
45. It is rather hard, due to the rapid evolution of the field to give an update reference on the subject. However, the following text gives a concise introduction of the basic principles involved in the physics of the neutron stars: S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars, The Physics of Compact Objects (Wiley-Interscience, New York, 1983).In addition an overview of the various possibilities regarding the behavior of the core magnetic fields can be found in J. Sauls, in Timing of Neutron Stars, edited by H. Ogelman and E. P. J. van der Heuvel (Kluwer, Dordrecht, 1989).
46. The idea that the core collapse and supernovas explosion may also be accompanied by a heavy accretion has been discussed long ago in the following: S. Colgate, Astrophys. J. 163, 221 (1971)
47. Y. B. Zeldovich, L. N. Ivanova, and D. K. Nadezhin, Astron. Zh. 49, 253 (1972) [Sov. Astron. 16, 209 (1972)].
48. A. Muslimov and D. Page, Astrophys. J. Lett. 440, L77 (1995)
49. Astrophys. J. Lett.A. MuslimovD. Page458, 347 (1995).
50. U. Geppert, D. Page, and T. Zannias, Astron. Astrophys. 345, 847 (1999)
51. U. GeppertD. PageT. ZanniasMem. Soc. Astron. Ital. 69, 4 (1998).
52. For a review of a few elementary properties of the vector calculus in orthogonal curvilinear coordinates, consult Ref. 20 above as well as Fundamental Formulas in Physics, edited by D. H. Menzel (Dover, New York, 1960), Vol. II.