A Pedagogical Discussion Concerning the Gravitational Energy Radiated by Keplerian Systems

📝 Abstract

We first discuss the use of dimensional arguments (and of the quadrupolar emission hypothesis) in the derivation of the gravitational power radiated on a circular orbit. Then, we show how to simply obtain the instantaneous power radiated on a general Keplerian orbit by approximating it locally by a circle. This allows recovering with a good precision, in the case of an ellipse, the highly non trivial dependence on the eccentricity of the average power given by general relativity. The whole approach is understandable by undergraduate students.

💡 Analysis

We first discuss the use of dimensional arguments (and of the quadrupolar emission hypothesis) in the derivation of the gravitational power radiated on a circular orbit. Then, we show how to simply obtain the instantaneous power radiated on a general Keplerian orbit by approximating it locally by a circle. This allows recovering with a good precision, in the case of an ellipse, the highly non trivial dependence on the eccentricity of the average power given by general relativity. The whole approach is understandable by undergraduate students.

📄 Content

1 A PEDAGOGICAL DISCUSSION CONCERNING THE GRAVITATIONAL ENERGY RADIATED BY KEPLERIAN SYSTEMS LAPTH-1314/09

Christian Bracco UMR Fizeau, Université de Nice-Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, Campus Valrose, F-06108 Nice Cedex and Syrte, CNRS, Observatoire de Paris, 61 avenue de l’Observatoire, F-75014 Paris

Jean-Pierre Provost INLN, Université de Nice-Sophia Antipolis, 1361 route des lucioles, Sophia Antipolis, F-06560 Valbonne

Pierre Salati Laboratoire d’Annecy-Le-Vieux de Physique Théorique LAPTH, Université de Savoie et CNRS, 9 Chemin de Bellevue, B.P. 110, F-74941 Annecy-Le-Vieux Cedex

We first discuss the use of dimensional arguments (and of the quadrupolar emission hypothesis) in the derivation of the gravitational power radiated on a circular orbit. Then, we show how to simply obtain the instantaneous power radiated on a general Keplerian orbit by approximating it locally by a circle. This allows recovering with a good precision, in the case of an ellipse, the highly non trivial dependence on the eccentricity of the average power given by general relativity. The whole approach is understandable by undergraduate students.

Keywords: gravitational waves, Keplerian orbits, PSR 1913+16, dimensional analysis, radius of curvature

PACS: 04.30.-w, 95.85.Sz

2 I. GRAVITATIONAL ENERGY RADIATED BY KEPLERIAN SYSTEMS; INTRODUCTION

Einstein built the general theory of relativity (GR) between 1907, when he formulated the first version of the equivalence principle, and November 1915, when he obtained his equations for the gravitational field. In GR, space-time is described by the metric tensor, whose components are identified with the gravitational potentials of the matter. In 1918, Einstein established that small perturbations of this tensor propagate at the speed of light and are generated by masses undergoing acceleration. These perturbations, which are called gravitational waves (GW) describe transverse (shear) deformations of space and are associated with the quadrupolar momentum (inertia momentum) of the source1,2 (whereas in electromagnetism the transverse polarization is a vectorial one and is generated in first approximation by the dipolar momentum of the source).

In the case of a binary system, the energy carried away by GW is lost by the system, hence a decrease of its orbital period. The first detection of that effect occurred in the 1980’s after the discovery in 19753 of the binary pulsar PSR 1913+16 by Hulse and Taylor, who were rewarded by the Nobel Prize in 1993.4 This system is composed of two neutron stars with 1.44 and 1.38 solar masses, orbiting with a short period of 7.8 h on Keplerian elliptical trajectories with a noticeable eccentricity 0.617. Thanks to many years of observations by Taylor and Weisberg, the data were accurate enough to show that the decrease of the orbital period is consistent, up to a precision of measurement of 0.4%, with the emission of GW predicted by GR.5 GW detection is an intense field of research, and huge detectors, such as the LIGO and VIRGO (or LISA in the future) interferometers6 have been built to detect GW as they reach the Earth.

3 The aim of this paper is to show that standard knowledge at undergraduate level can be used to calculate the GW power radiated by a celestial body such as PSR 1913+16, and to discuss its dependence on the various orbital parameters (including eccentricity). In section II, we use dimensional arguments to obtain (up to a constant) the gravitational power radiated on a circular orbit under the assumption of a quadrupolar emission. The same analysis, applied to the dipolar and quadrupolar electromagnetic radiations leads to an interesting comparison. Of course, this dimensional approach does not apply to a general Keplerian orbit because of the eccentricity, which is an adimensional parameter. In section III, we show how to derive very simply the instantaneous power of radiation on such an orbit, by introducing the local radius of curvature (which enters the well known expression of the normal acceleration). Finally in section IV, we discuss the case of a Keplerian ellipse and calculate the mean radiated power. The dependence of this power with respect to the eccentricity of the orbit agrees with a precision of order 1% with the formula7 of GR which has been confronted to the observations.

II. DIMENSIONAL CALCULUS AND THE CASE OF A CIRCULAR ORBIT

In the center of mass frame of a binary system, let ( ) 1 2 1 2 = m m m +m µ be the reduced mass, R be the radius of its circular orbit and T = 2π ω be its orbital period. The power circ P of the gravitational radiation is expected to depend on µ, R and ω, as well as on the coupling constant of gravitation G and on the speed of light c. It is impossible to obtain a unique formula for circ P from a dimensional argument with these five paramet

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