New Hamiltonian expansions adapted to the Trojan problem

A number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez and Locatelli in MNRAS 453(2):2177, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem. In Páez and Efthymiopoulos (2015), the motion of a Trojan body was studied in the context of the planar Elliptic Restricted Three Body or the planar Restricted Multi-Planet Problem. It was shown that the Hamiltonian admits a generic decomposition \(H=H_b+H_{sec}\). The term \(H_b\), called the basic Hamiltonian, is a model of two degrees of freedom characterizing the short-period and synodic motions of a Trojan body. Also, it yields a constant ‘proper eccentricity’ allowing to define a third secular frequency connected to the body’s perihelion precession. \(H_{sec}\) contains all remaining secular perturbations due to the primary or to additional perturbing bodies. Here, we first investigate up to what extent the decomposition \(H=H_b+H_{sec}\) provides a meaningful model. To this end, we produce numerical examples of surfaces of section under \(H_b\) and compare with those of the full model. We also discuss how secular perturbations alter the dynamics under \(H_b\). Secondly, we explore the normal form approach introduced in Páez and Locatelli (2015) in order to find an ‘averaged over the fast angle’ model derived from \(H_b\), circumventing the problem of the series’ limited convergence due to the collision singularity at the 1:1 MMR. Finally, using this averaged model, we compute semi-analytically the position of the most important secondary resonances and compare the results with those found by numerical stability maps in specific examples. We find a very good agreement between semi-analytical and numerical results in a domain whose border coincides with the transition to large-scale chaotic Trojan motions.     

Secondary resonances and the boundary of effective stability of Trojan motions

One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced ‘basic Hamiltonian model’ \(H_\mathrm{b}\) for Trojan dynamics (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez et al. in Celest Mech Dyn Astron 126:519, 2016): we show that the inner border of the secondary resonance of lowermost order, as defined by \(H_\mathrm{b}\), provides a good estimation of the region in phase space for which the orbits remain regular regardless of the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an ‘asymmetric expansion’ of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed.     

Dynamics of “jumping” Trojans: a perturbative treatment

The term “jumping” Trojan was introduced by Tsiganis et al. (Astron Astrophys 354:1091–1100, 2000) in their studies of long-term dynamics exhibited by the asteroid (1868) Thersites, which had been observed to jump from librations around \(L_4\) to librations around \(L_5\). Another example of a “jumping” Trojan was found by Connors et al. (Nature 475:481–483, 2011): librations of the asteroid 2010 TK7 around the Earth’s libration point \(L_4\) preceded by its librations around \(L_5\). We explore the dynamics of “jumping” Trojans under the scope of the restricted planar elliptical three-body problem. Via double numerical averaging we construct evolutionary equations, which allow analyzing transitions between different regimes of orbital motion.     

Free time minimizers for the three-body problem

Free time minimizers of the action (called “semi-static” solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120–131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton–Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton’s three-body problem which is asymptotic to Lagrange’s parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange’s solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209–227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler’s central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.     

The eccentric Kozai–Lidov effect as a resonance phenomenon

Exploring weakly perturbed Keplerian motion within the restricted three-body problem, Lidov (Planet Space Sci 9:719–759, 1962) and, independently, Kozai (Astron J 67:591–598, 1962) discovered coupled oscillations of eccentricity and inclination (the KL cycles). Their classical studies were based on an integrable model of the secular evolution, obtained by double averaging of the disturbing function approximated with its first non-trivial term. This was the quadrupole term in the series expansion with respect to the ratio of the semimajor axis of the disturbed body to that of the disturbing body. If the next (octupole) term is kept in the expression for the disturbing function, long-term modulation of the KL cycles can be established (Ford et al. in Astrophys J 535:385–401, 2000; Naoz et al. in Nature 473:187–189, 2011; Katz et al. in Phys Rev Lett 107:181101, 2011). Specifically, flips between the prograde and retrograde orbits become possible. Since such flips are observed only when the perturber has a nonzero eccentricity, the term “eccentric Kozai–Lidov effect” (or EKL effect) was proposed by Lithwick and Naoz (Astrophys J 742:94, 2011) to specify such behavior. We demonstrate that the EKL effect can be interpreted as a resonance phenomenon. To this end, we write down the equations of motion in terms of “action-angle” variables emerging in the integrable Kozai–Lidov model. It turns out that for some initial values the resonance is degenerate and the usual “pendulum” approximation is insufficient to describe the evolution of the resonance phase. Analysis of the related bifurcations allows us to estimate the typical time between the successive flips for different parts of the phase space.     

Tidal synchronization of close-in satellites and exoplanets. III. Tidal dissipation revisited and application to Enceladus

This paper deals with a new formulation of the creep tide theory (Ferraz-Mello in Celest Mech Dyn Astron 116:109, 2013—Paper I) and with the tidal dissipation predicted by the theory in the case of stiff bodies whose rotation is not synchronous but is oscillating around the synchronous state with a period equal to the orbital period. We show that the tidally forced libration influences the amount of energy dissipated in the body and the average perturbation of the orbital elements. This influence depends on the libration amplitude and is generally neglected in the study of planetary satellites. However, they may be responsible for a 27% increase in the dissipation of Enceladus. The relaxation factor necessary to explain the observed dissipation of Enceladus (\(\gamma =1.2{-}3.8\times 10^{-7}\ \mathrm{s}^{-1}\)) has the expected order of magnitude for planetary satellites and corresponds to the viscosity \(0.6{-}1.9 \times 10^{14}\) Pa s, which is in reasonable agreement with the value recently estimated by Efroimsky (Icarus 300:223, 2018) (\(0.24 \times 10^{14}\) Pa s) and with the value adopted by Roberts and Nimmo (Icarus 194:675, 2008) for the viscosity of the ice shell (\(10^{13}{-}10^{14}\) Pa s). For comparison purposes, the results are extended also to the case of Mimas and are consistent with the negligible dissipation and the absence of observed tectonic activity. The corrections of some mistakes and typos of paper II (Ferraz-Mello in Celest Mech Dyn Astron 122:359, 2015) are included at the end of the paper.     

The fallacy of Oppenheimer Snyder collapse: no general relativistic collapse at all,no black hole,no physical singularity

By applying Birkhoff’s theorem to the problem of the general relativistic collapse of a uniform density dust, we directly show that the density of the dust ρ=0 even when its proper number density n would be assumed to be finite! The physical reason behind this exact result can be traced back to the observation of Arnowitt et al. (Phys. Rev. Lett. 4: 375, 1960) that the gravitational mass of a neutral point particle is zero: m=0. And since, a dust is a mere collection of neutral point particles, unlike a continuous hydrodynamic fluid, its density ρ=mn=0. It is nonetheless found that for k=?1, a homogeneous dust can collapse and expand special relativistically in the fashion of a Milne universe. Thus, in reality, general relativistic homogeneous dust collapse does not lead to the formation of any black hole in conformity of many previous studies (Logunov et al., Phys. Part. Nucl. 37: 317, 2006; Kiselev et al., Theor. Math. Phys. 164: 972, 2010; Mitra, J. Math. Phys. 50: 042502, 2009a; Suggett, J. Phys. A 12: 375 1979b). Interestingly, this result is in agreement with the intuition of Oppenheimer and Snyder (Phys. Rev. 56: 456, 1939) too:“Physically such a singularity would mean that the expressions used for the energy-momentum tensor does not take into account some essential physical fact which would really smooth the singularity out. Further, a star in its early stages of development would not possess a singular density or pressure, it is impossible for a singularity to develop in a finite time.”     

Stark Broadening Parameters for Neutral Oxygen Spectral Lines

Stark broadening parameters for nine neutral oxygen (O I) lines have been determined within the impact approximation and the semiclassical perturbation method. The atomic data have been taken from the TOPbase and NIST atomic databases. The electron and proton Stark widths and shifts and ion broadening parameter values for these O I lines have been calculated for electron density of 10 ¹⁶ cm ^?3 and for 4 different electron temperatures in the range of 5000 K to 40000 K. These Stark broadening parameters are compared with our previous results (Ben Nessib, N. et al. 1996, Physica Scripta, 54, 603–613), where we calculated Stark broadening parameters for only four O I spectral lines and where Stark widths and shifts were compared with experimental and theoretical data available in the literature. In the present paper, we have also compared our results with the Griem’s book (Griem, H. R. 1974, Spectral line broadening by plasmas) and VALD (Ryabchikova, T. et al. 2015, Physica Scripta, 90, 054005) values.     

Halo orbit of regularized circular restricted three-body problem with radiation pressure and oblateness

In this paper, computation of the halo orbit for the KS-regularized photogravitational circular restricted three-body problem is carried out. This work extends the idea of Srivastava et al. (Astrophys. Space Sci. 362: 49, 2017) which only concentrated on the (i) regularization of the 3D-governing equations of motion, and (ii) validation of the modeling for small out-of-plane amplitude (\(A_z =110000\) km) assuming the third-order analytical approximation as an initial guess with and without differential correction. This motivated us to compute the halo orbits for the large out-of-plane amplitudes and to study their stability analysis for the regularized motion. The stability indices are described as a function of out-of-plane amplitude, mass reduction factor and oblateness coefficient. Three different Sun–planet systems: the Sun–Earth, Sun–Mars and the Sun–Jupiter are chosen in this study. Stable halo orbits do not exist around the \(L_{1}\) point, however, around the \(L_{2}\) point stable halo orbits are found for the considered systems.     

The rectilinear three-body problem as a basis for studying highly eccentric systems

The rectilinear elliptic restricted three-body problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity \(e'=1\), but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (AIAA J 7:1003–1009, 1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter \(\mu =0.5\) (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke’s computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to \(\mu \) and \(e'<1\). Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.     

Dynamics of Large-Scale Solar-Wind Streams Obtained by the Double Superposed Epoch Analysis: 2. Comparisons of CIRs <Emphasis Type="Italic">vs</Emphasis>. Sheaths and MCs <Emphasis Type="Italic">vs</Emphasis>. Ejecta

This work is a continuation of our previous article (Yermolaev et al. in J. Geophys. Res. 120, 7094, 2015), which describes the average temporal profiles of interplanetary plasma and field parameters in large-scale solar-wind (SW) streams: corotating interaction regions (CIRs), interplanetary coronal mass ejections (ICMEs including both magnetic clouds (MCs) and ejecta), and sheaths as well as interplanetary shocks (ISs). As in the previous article, we use the data of the OMNI database, our catalog of large-scale solar-wind phenomena during 1976?–?2000 (Yermolaev et al. in Cosmic Res., 47, 2, 81, 2009) and the method of double superposed epoch analysis (Yermolaev et al. in Ann. Geophys., 28, 2177, 2010a). We rescale the duration of all types of structures in such a way that the beginnings and endings for all of them coincide. We present new detailed results comparing pair phenomena: 1) both types of compression regions (i.e. CIRs vs. sheaths) and 2) both types of ICMEs (MCs vs. ejecta). The obtained data allow us to suggest that the formation of the two types of compression regions responds to the same physical mechanism, regardless of the type of piston (high-speed stream (HSS) or ICME); the differences are connected to the geometry (i.e. the angle between the speed gradient in front of the piston and the satellite trajectory) and the jumps in speed at the edges of the compression regions. In our opinion, one of the possible reasons behind the observed differences in the parameters in MCs and ejecta is that when ejecta are observed, the satellite passes farther from the nose of the area of ICME than when MCs are observed.     

Prospects of detecting fast radio bursts using Indian radio telescopes

Fast Radio Bursts (FRBs) are short duration highly energetic dispersed radio pulses. We developed a generic formalism (Bera et al. 2016, MNRAS, 457, 2530) to estimate the FRB detection rate for any radio telescope with given parameters. By using this model, we estimated the FRB detection rate for two Indian radio telescope; the Ooty Wide Field Array (OWFA) (Bhattacharyya et al. 2017, J. Astrophys. Astr., 38, 17) and the upgraded Giant Metrewave Radio Telescope (uGMRT) (Bhattacharyya et al. 2018, J. Astrophys. Astr.) with three beam-forming modes. Here, we summarize these two works. We considered the energy spectrum of FRBs as a power law and the energy distribution of FRBs as a Dirac delta function and a Schechter luminosity function. We also considered two scattering models proposed by Bhat et al. (2004, Astrophys. J. Suppl. Series, 206, 1) and Macquart & Koay (2013, ApJ, 776, 125) for these works and we consider FRB pulse without scattering as a special case. We found that the future prospects of detecting FRBs by using these two Indian radio telescopes is good. They are capable to detect a significant number of FRBs per day. According to our prediction, we can detect \(\sim 10^5{-}10^8\), \(\sim 10^3{-}10^6\) and \(\sim 10^5{-}10^7\) FRBs per day by using OWFA, commensal systems of GMRT and uGMRT respectively. Even a non detection of the predicted events will be very useful in constraining FRB properties.     

Using GALEX-SDSS-PanSTARRS-HST-Gaia to understand post-AGB evolution

Hot WDs in binary systems with a less evolved star are particularly invaluable astrophysical probes, the unevolved companion enabling better derivation of distance and age than is usually possible for post-AGB objects, and therefore also of their radius and luminosity. But hot white dwarfs (WD) are elusive at all wavelengths except the UV (Bianchi et al. 2011a). From our GALEX UV source catalogs (Bianchi et al. 2011a,b, 2014, 2017) matched to SDSS, we identified thousands of candidate hot WDs including WDs in binary systems consisting of a hot WD and a companion of spectral type from A to M. The identification and preliminary characterization of the stellar parameters is based on the analysis of the photometric SED from far-UV to z-band.We have observed subsamples of the UV-selected WDs with the Hubble Space Telescope (HST) to better characterize their stellar parameters. We obtained (1) UV spectroscopy with STIS and analyzed the UV spectra together with optical SDSS spectra, and (2) multi-band imaging with WFC3 (\(0.04^{\prime\prime}/\mbox{pixel}\)) to measure angular separation and individual SEDs of the pair’s components in binary systems. In our HST/WFC3 sample of 59 hot-WD binaries with late-type companions, we found that at least a dozen have possibly evolved without exchanging mass. The UV STIS spectroscopy led to the revision of previous results based on optical spectra only, because of the often undetectable or unquantifiable contribution of the hot component to the optical fluxes.     

Variations in the Solar Coronal Rotation with Altitude – Revisited

Here we report an in-depth reanalysis of an article by Vats et al. (Astrophys. J. 548, L87, 2001) that was based on measurements of differential rotation with altitude as a function of observing frequencies (as lower and higher frequencies indicate higher and lower heights, respectively) in the solar corona. The radial differential rotation of the solar corona is estimated from daily measurements of the disc-integrated solar radio flux at 11 frequencies: 275, 405, 670, 810, 925, 1080, 1215, 1350, 1620, 1755, and 2800 MHz. We use the same data as were used in Vats et al. (2001), but instead of the twelfth maxima of autocorrelograms used there, we use the first secondary maximum to derive the synodic rotation period. We estimate synodic rotation by Gaussian fit of the first secondary maximum. Vats et al. (2001) reported that the sidereal rotation period increases with increasing frequency. The variation found by them was from 23.6 to 24.15 days in this frequency range, with a difference of only 0.55 days. The present study finds that the sidereal rotation period increases with decreasing frequency. The variation range is from 24.4 to 22.5 days, and the difference is about three times larger (1.9 days). However, both studies give a similar rotation period at 925 MHz. In Vats et al. (2001) the Pearson’s factor with trend line was 0.86, whereas present analysis obtained a \({\sim}\,0.97\) Pearson’s factor with the trend line. Our study shows that the solar corona rotates more slowly at higher altitudes, which contradicts the findings reported in Vats et al. (2001).     

Einstein–Maxwell Field Equation in Isotropic Coordinates: An Application to Modeling Superdense Star

We present a charged analogue of Pant et al. (2010, Astrophys. Space Sci., 330, 353) solution of the general relativistic field equations in isotropic coordinates by using simple form of electric intensity E that involve charge parameter K. Our solution is well behaved in all respects for all values of X lying in the range 0 <X≤ 0.11, K lying in the range 4 <K≤ 6.2 and Schwarzschild compactness parameter u lying in the range 0 <u≤ 0.247. Since our solution is well behaved for wide ranges of the parameters, we can model many different types of ultra-cold compact stars like quark stars and neutron stars. We have shown that corresponding to X = 0.077 and K = 6.13 for which u = 0.2051 and by assuming surface density ρ _b =4.6888×10¹⁴ g cm ^?3 the mass and radius are found to be 1.509M _⊙, 10.906 km respectively which match with the observed values of mass 1.51M _⊙ and radius 10.90 km of the quark star XTE J1739-217. The well behaved class of relativistic stellar models obtained in this work might have astrophysical significance in the study of more realistic internal structures of compact stars.     

On the photo-gravitational restricted four-body problem with variable mass

This paper deals with the photo-gravitational restricted four-body problem (PR4BP) with variable mass. Following the procedure given by Gascheau (C. R. 16:393–394, 1843) and Routh (Proc. Lond. Math. Soc. 6:86–97, 1875), the conditions of linear stability of Lagrange triangle solution in the PR4BP are determined. The three radiating primaries having masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) in an equilateral triangle with \(m_{2}=m_{3}\) will be stable as long as they satisfy the linear stability condition of the Lagrangian triangle solution. We have derived the equations of motion of the mentioned problem and observed that there exist eight libration points for a fixed value of parameters \(\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )\), \(\alpha\) (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), \(0\leq\alpha\leq2.2\)), the mass parameter \(\mu=0.005\) and radiation parameters \(q_{i}, (0< q_{i}\leq1, i=1, 2, 3)\). All the libration points are non-collinear if \(q_{2}\neq q_{3}\). It has been observed that the collinear and out-of-plane libration points also exist for \(q_{2}=q_{3}\). In all the cases, each libration point is found to be unstable. Further, zero velocity curves (ZVCs) and Newton–Raphson basins of attraction are also discussed.     

Transition between general relativity and quantum gravity using quark and strange quark matter with some kinematical test

Contemporary piece of writing devotes to the investigation of plane symmetric cosmological model with quark and strange quark matter in the deformations of the Einstein’s theory of General Relativity (GR). At small or large scales (ultraviolet or infrared gravity), deformations of the Einstein’s theory could provide a better handling of cosmic acceleration without magnetism (along with singularities). In particular, a proper deformation of GR in the ultraviolet regime could play the role of describing the transition between GR and quantum gravity. As a matter of fact, although with a different purpose in mind, it was Einstein himself who proposed in the 30’s the reformulation of GR by taking the field of orthonormal frames or tetrads as the dynamical variable instead of the metric tensor (Einstein, Phys. Math. Kl 217, 401, 1928). As per the observation, pressure and energy density of the model approaches the bag constant in negative and positive ways at \(t\rightarrow \infty \), i.e. \(p\rightarrow -B_c \) and \(\rho \rightarrow B_c \), the negative pressure due to the Dark Energy (DE) in the context of accelerated expansion of the universe. So the strange quark matter gives an idea of existence of dark energy in the universe and supports the observations of the SNe-I (Riess et al., Astron. J. 116,1009, 1998; Perlmutter et al., Astrophys. J. 517, 565, 1999). Also these results agree with the study of Aktas and Aygun (Chinese J. Phys. 55, 71, 2017) and Sahoo et al. (New. Astron. 60, 80, 2018).     

Inclined asymmetric librations in exterior resonances

Librational motion in Celestial Mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or \(\pi \), the libration is called asymmetric. In the context of the planar circular restricted three-body problem, asymmetric librations have been identified for the exterior mean motion resonances (MMRs) 1:2, 1:3, etc., as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the three-dimensional space. We employ the computational approach of Markellos (Mon Not R Astron Soc 184:273–281,  https://doi.org/10.1093/mnras/184.2.273, 1978) and compute families of asymmetric periodic orbits and their stability. Stable asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun–Neptune–TNO system (TNO = trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2 resonant, inclined asymmetric librations. For the stable 1:2 TNO librators, we find that their libration seems to be related to the vertically stable planar asymmetric orbits of our model, rather than the three-dimensional ones found in the present study.     

Cosmic horizon for GeV sources and photon-photon scattering

Photon-photon scattering of gamma-rays on the cosmic microwave background has been studied using the low energy approximation of the total cross section by Zdziarski and Svensson (Astrophys. J. 344:551, 1989), Svensson and Zdziarski (Astrophys. J. 349:415, 1990). Here, the cosmic horizon due to photon-photon scattering is accurately determined using the exact cross section and we find that photon-photon scattering dominates over the pair production at energies smaller than 1.68 GeV and at redshifts larger than 180.     

The effects of deformation inertia (kinetic energy) in the orbital and spin evolution of close-in bodies

The purpose of this work is to evaluate the effect of deformation inertia on tide dynamics, particularly within the context of the tide response equations proposed independently by Boué et al. (Celest Mech Dyn Astron 126:31–60, 2016) and Ragazzo and Ruiz (Celest Mech Dyn Astron 128(1):19–59, 2017). The singular limit as the inertia tends to zero is analyzed, and equations for the small inertia regime are proposed. The analysis of Love numbers shows that, independently of the rheology, deformation inertia can be neglected if the tide-forcing frequency is much smaller than the frequency of small oscillations of an ideal body made of a perfect (inviscid) fluid with the same inertial and gravitational properties of the original body. Finally, numerical integration of the full set of equations, which couples tide, spin and orbit, is used to evaluate the effect of inertia on the overall motion. The results are consistent with those obtained from the Love number analysis. The conclusion is that, from the point of view of orbital evolution of celestial bodies, deformation inertia can be safely neglected. (Exceptions may occur when a higher-order harmonic of the tide forcing has a high amplitude.)