Orbital effects of a monochromatic plane gravitational wave with ultra-low frequency incident on a gravitationally bound two-body system

SOR-ASTRO Orbital effects of a monochromatic plane gravitational wave with ultra-low frequency incident on a gravitationally bound two-body system Lorenzo Iorio* Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.), Rome, Italy *Corresponding author’s e-mail address: Published online: April 28, 2014 (version 1) Cite as: Iorio L., ScienceOpen Research 2014 (DOI: 10.14293/A2199-1006.01.SOR-ASTRO.WXWIL.v1) Reviewing status: Please note that this article is under continuous review. For the current reviewing status and the latest referee’s comments please click here or scan the QR code at the end of this article. Primary discipline: Astronomy & Astrophysics Associated discipline: Cosmology & Extragalactic astrophysics, General relativity & Quantum cosmology Keywords: Relativity and gravitation, Gravitational waves, Celestial mechanics, Orbit determination and improvement PACS: 95.30.Sf, 04.30.-w, 95.10.Ce, 95.10.Eg ABSTRACT We analytically compute the long-term orbital variations of a test particle orbiting a central body acted upon by an incident monochromatic plane gravitational wave. We assume that the characteristic size of the perturbed two-body system is much smaller than the wavelength of the wave. Moreover, we also suppose that the wave’s frequency n g is much smaller than the particle’s b nor on the orbital configuration orbital one nb. We make neither a priori assumptions about the direction of the wavevector k of the particle. While the semi-major axis a is left unaffected, the eccentricity e, the inclination I, the longitude of the ascending node Ω, the longitude of pericenter ϖ and the mean anomaly M undergo non-vanishing long-term changes of the form dW=dt ¼ FðKij ; e; I; X; xÞ; W ¼ e; I; X; -; M, where Kij ; i; j ¼ 1; 2; 3 are the coefficients of the tidal matrix K. Thus, in addition to the variations of its orientation in space, the shape of the orbit would be altered as well. Strictly speaking, such effects are not secular trends because of the slow modulation introduced by K and by the orbital elements themselves: they exhibit peculiar long-term temporal patterns which would be potentially of help for their detection in multidecadal analyses of extended data records of planetary observations of various kinds. In particular, they could be useful in performing independent tests of the inflation-driven ultra-low gravitational waves whose imprint may have been indirectly detected in the Cosmic Microwave Background by the Earthbased experiment BICEP2. Our calculation holds, in general, for any gravitationally bound two-body system whose orbital frequency nb is much larger than the frequency n g of the external wave, like, e.g., extrasolar planets and the stars orbiting the Galactic black hole. It is also valid for a generic perturbation of tidal type with constant coefficients over timescales of the order of the orbital period of the perturbed particle. 1. INTRODUCTION Gravitational waves [1, 2] are a key theoretical prediction of the general theory of relativity (GTR). Indeed, since GTR relies upon the Lorentz invariance, which carries with it the concept of a limiting speed for physical interactions, the existence of gravitational waves is a natural consequence of it. A direct measurement of them is still lacking; see, e.g., Cerdonio [3]; Giazotto [4]; Fairhurst et al. [5] for recent reviews of the status of gravitational wave detection. To date, only indirect evidences of their existence have been inferred from the orbital decay rate of the binary pulsar PSR B1913+16 [6] and, more recently, from the detection [7] of B – mode polarization at degree angular scales in the Cosmic Microwave Background (CMB) by the ground-based experiment BICEP2 [8] at the South Pole. The consequences of detecting gravitational waves for physics, astrophysics, and cosmology would be certainly remarkable; see, e.g., Sathyaprakash and Schutz [9]. The role of a direct detection of the gravitational waves for GTR and extended theories of gravity was pointed out by Corda [10, 11]. The gravitational wave spectrum covers an interval of about 18 orders of magnitude in wavelengths, encompassing a very broad range of physics and astrophysics [12]. The frequencies in the range 101104 Hz are the targets of several ground-based detectors like, LIGO [13, 14], VIRGO [15–17], TAMA [18–20], GEO [21, 22], etc. Typical sources of such high-frequency waves are coalescing binary systems hosting stellar-sized compact objects like neutron stars and/or black holes; see, e.g., Section 6.1 of 1 SOR-ASTRO L. Iorio: Orbital effects of a monochromatic plane gravitational wave on a gravitationally bound two-body system Flanagan and Hughes [1]. The space-based LISA mission1 [24–26], now evolved into eLISA2 [27], aimed to detect gravitational waves in the frequency range 1051 Hz, while accurate timing measurements of pulsars may detect signals in the range 109107 Hz [28–30]. Section 6.2 of Flanagan and Hughes [1] yields an overview of typical LISA sources: they include, e.g., equal mass binaries, in which the member black holes are of roughly equal mass (~105108 M⊙), and extreme mass ratio binaries made by white dwarfs, neutron stars and 10100M⊙ black holes captured by 105107 M⊙ black holes, located at quite large distances. As far as the very low frequency band is concerned (109107 Hz), refer Section 6.3 of Flanagan and Hughes [1]. It may be due to several unresolved coalescing massive black holes; binaries which are either too massive to emit in the LISA band, or else are in-spiralling and will finally merge in several centuries or millennia. In this paper, we will analytically work out the long-term orbital variations of all the six standard Keplerian orbital elements of a solar system planet due to the action of an externally incident monochromatic plane gravitational wave in the green–black3 part of the spectrum, i.e. with frequency n g 5107  1010 Hz. Such kinds of gravitational waves are important since they carry information about how galaxies and black holes co-evolved over the history of the universe [31, 32], the early universe and related exotic physical processes like, e.g., inflation and cosmic strings, and possible physics beyond the standard model of particles and fields [33–37]. In particular, a primordial background of ultra-low frequency stochastic gravitational waves with a characteristic spectral shape should be produced due to the coupling of the gravitational field with the exponential expansion driven by the inflation [34, 35, 38–41]. Concerning the possible existence of a background of gravitational waves dating back to the origin of the universe, see, e.g., Weber [42]; Wheeler [43]; Zel’dovich and Novikov [44]; Carr [45], and the discussion in Mashhoon et al. [46]. The effects of incident gravitational waves on the orbital motion of gravitationally bound systems were inspected by several authors with a variety of approaches and approximations per (...truncated)