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.     

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.     

A family of stacked central configurations in the planar five-body problem

We study planar central configurations of the five-body problem where three bodies, \(m_1, m_2\) and \(m_3\), are collinear and ordered from left to right, while the other two, \(m_4\) and \(m_5\), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with \(m_1=m_3\), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments \(m_4m_5\) and \(m_1m_3\) do not intersect.     

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.     

Fast numerics for the spin orbit equation with realistic tidal dissipation and constant eccentricity

We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is \(C^1\) in the velocity. Such an ODE arises as a model of spin–orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on timescales of the order of \(10^6\)–\(10^7\) years. The proposed algorithm is based on the high-order Euler method which was described in Bartuccelli et al. (Celest Mech Dyn Astron 121(3):233–260, 2015), and it requires computer algebra to set up the code for its implementation. The payoff is an overall increase in speed by a factor of about 7.5 compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed.     

Viscoelastic tides: models for use in Celestial Mechanics

In this note, by using Smale’s \(\alpha \)-theorem on the convergence of Newton’s method, the \(\alpha \)-sets of convergence of some starters of solving the elliptic Kepler’s equation are derived. For each starter we compute the exact \(\alpha \)-set in the eccentricity-main anomaly \((e,M)\in [0,1)\times [0,\pi ]\), showing that these sets are larger than those derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014). Further, new convergence tests based on the Newton–Kantorowitch theorem are given comparing with the derived from Smale’s \(\alpha \)-test.     

An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

In this note a study of the convergence properties of some starters \( E_0 = E_0(e,M)\) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter \( E_0(e,M)\) a set of values \( (e,M) \in [0, 1) \times [0, \pi ]\) that lead to the q-convergence in the sense that Newton’s sequence \( (E_n)_{n \ge 0}\) generated from \( E_0 = E_0(e,M)\) is well defined, converges to the exact solution \(E^* = E^*(e,M)\) of KE and further \( \vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert \) holds for all \( n \ge 0\). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s \(\alpha \)-test with \(q=1/2\). Also since in KE the convergence rate of Newton’s method tends to zero as \( e \rightarrow 0\), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with \( q = e^k \; \widetilde{q} \) for all \( e \in [0,1)\) and a fixed \( \widetilde{q} \le 1\). Some remarks on the use of this theorem to derive a priori estimates of the error \( \vert E_n - E^* \vert \) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.     

Study and application of the resonant secular dynamics beyond Neptune

We use a secular representation to describe the long-term dynamics of transneptunian objects in mean-motion resonance with Neptune. The model applied is thoroughly described in Saillenfest et al. (Celest Mech Dyn Astron, doi: 10.1007/s10569-016-9700-5, 2016). The parameter space is systematically explored, showing that the secular trajectories depend little on the resonance order. High-amplitude oscillations of the perihelion distance are reported and localised in the space of the orbital parameters. In particular, we show that a large perihelion distance is not a sufficient criterion to declare that an object is detached from the planets. Such a mechanism, though, is found unable to explain the orbits of Sedna or \(2012\text {VP}_{113}\), which are insufficiently inclined (considering their high perihelion distance) to be possibly driven by such a resonant dynamics. The secular representation highlights the existence of a high-perihelion accumulation zone due to resonances of type 1:k with Neptune. That region is found to be located roughly at \(a\in [100;300]\) AU, \(q\in [50;70]\) AU and \(I\in [30;50]^{\circ }\). In addition to the flux of objects directly coming from the Scattered Disc, numerical simulations show that the Oort Cloud is also a substantial source for such objects. Naturally, as that mechanism relies on fragile captures in high-order resonances, our conclusions break down in the case of a significant external perturber. The detection of such a reservoir could thus be an observational constraint to probe the external Solar System.     

Improved Determination of the Location of the Temperature Maximum in the Corona

The most used method to calculate the coronal electron temperature [\(T_{\mathrm{e}} (r)\)] from a coronal density distribution [\(n_{\mathrm{e}} (r)\)] is the scale-height method (SHM). We introduce a novel method that is a generalization of a method introduced by Alfvén (Ark. Mat. Astron. Fys. 27, 1, 1941) to calculate \(T_{\mathrm{e}}(r)\) for a corona in hydrostatic equilibrium: the “HST” method. All of the methods discussed here require given electron-density distributions [\(n_{\mathrm{e}} (r)\)] which can be derived from white-light (WL) eclipse observations. The new “DYN” method determines the unique solution of \(T_{\mathrm{e}}(r)\) for which \(T_{\mathrm{e}}(r \rightarrow \infty) \rightarrow 0\) when the solar corona expands radially as realized in hydrodynamical solar-wind models. The applications of the SHM method and DYN method give comparable distributions for \(T_{\mathrm{e}}(r)\). Both have a maximum [\(T_{\max}\)] whose value ranges between 1?–?3 MK. However, the peak of temperature is located at a different altitude in both cases. Close to the Sun where the expansion velocity is subsonic (\(r < 1.3\,\mathrm{R}_{\odot}\)) the DYN method gives the same results as the HST method. The effects of the other free parameters on the DYN temperature distribution are presented in the last part of this study. Our DYN method is a new tool to evaluate the range of altitudes where the heating rate is maximum in the solar corona when the electron-density distribution is obtained from WL coronal observations.     

Secular resonances between bodies on close orbits II: prograde and retrograde orbits for irregular satellites

In extending the analysis of the four secular resonances between close orbits in Li and Christou (Celest Mech Dyn Astron 125:133–160, 2016) (Paper I), we generalise the semianalytical model so that it applies to both prograde and retrograde orbits with a one-to-one map between the resonances in the two regimes. We propose the general form of the critical angle to be a linear combination of apsidal and nodal differences between the two orbits \( b_1 \Delta \varpi + b_2 \Delta \varOmega \), forming a collection of secular resonances in which the ones studied in Paper I are among the strongest. Test of the model in the orbital vicinity of massive satellites with physical and orbital parameters similar to those of the irregular satellites Himalia at Jupiter and Phoebe at Saturn shows that \({>}20\) and \({>}40\%\) of phase space is affected by these resonances, respectively. The survivability of the resonances is confirmed using numerical integration of the full Newtonian equations of motion. We observe that the lowest order resonances with \(b_1+|b_2|\le 3\) persist, while even higher-order resonances, up to \(b_1+|b_2|\ge 7\), survive. Depending on the mass, between 10 and 60% of the integrated test particles are captured in these secular resonances, in agreement with the phase space analysis in the semianalytical model.     

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.     

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.)     

Relating Solar Energetic Particle Event Fluences to Peak Intensities

Recently we (Kahler and Ling, Solar Phys.292, 59, 2017: KL) have shown that time–intensity profiles [\(I(t)\)] of 14 large solar energetic particle (SEP) events can be fitted with a simple two-parameter fit, the modified Weibull function, which is characterized by shape and scaling parameters [\(\alpha\) and \(\beta\)]. We now look for a simple correlation between an event peak energy intensity [\(I_{\mathrm{p}}\)] and the time integral of \(I(t)\) over the event duration: the fluence [\(F\)]. We first ask how the ratio of \(F/I_{\mathrm{p}}\) varies for the fits of the 14 KL events and then examine that ratio for three separate published statistical studies of SEP events in which both \(F\) and \(I_{\mathrm{p}}\) were measured for comparisons of those parameters with various solar-flare and coronal mass ejection (CME) parameters. The three studies included SEP energies from a 4?–?13 MeV band to \(E > 100~\mbox{MeV}\). Within each group of SEP events, we find a very robust correlation (\(\mathrm{CC} > 0.90\)) in log–log plots of \(F\)versus\(I_{\mathrm{p}}\) over four decades of \(I_{\mathrm{p}}\). The ratio increases from western to eastern longitudes. From the value of \(I_{\mathrm{p}}\) for a given event, \(F\) can be estimated to within a standard deviation of a factor of \({\leq}\,2\). Log–log plots of two studies are consistent with slopes of unity, but the third study shows plot slopes of \({<}\,1\) and decreasing with increasing energy for their four energy ranges from \(E > 10~\mbox{MeV}\) to \({>}\,100~\mbox{MeV}\). This difference is not explained.     

Modelling the rotation period distribution of M dwarfs in the <Emphasis Type="Italic">Kepler</Emphasis> field

McQuillan et al. (Mon. Not. R. Astron. Soc. 432:1203, 2013) presented 1570 periods \(P\) of M dwarf stars in the field of view of the Kepler telescope. It is expected that most of these reflect rotation periods, due to starspots. It is shown here that the data can be modelled as a mixture of four subpopulations, three of which are overlapping log-normal distributions. The fourth subpopulation has a power law distribution, with \(P^{-1/2}\). It is also demonstrated that the bulk of the longer periods, representing the two major subpopulations, could be drawn from a single subpopulation, but with a period-dependent probability of observing half the true period.     

Recursive computation of mutual potential between two polyhedra

Recursive computation of mutual potential, force, and torque between two polyhedra is studied. Based on formulations by Werner and Scheeres (Celest Mech Dyn Astron 91:337–349, 2005) and Fahnestock and Scheeres (Celest Mech Dyn Astron 96:317–339, 2006) who applied the Legendre polynomial expansion to gravity interactions and expressed each order term by a shape-dependent part and a shape-independent part, this paper generalizes the computation of each order term, giving recursive relations of the shape-dependent part. To consider the potential, force, and torque, we introduce three tensors. This method is applicable to any multi-body systems. Finally, we implement this recursive computation to simulate the dynamics of a two rigid-body system that consists of two equal-sized parallelepipeds.     

Short timescale UV variability study in NGC 4151 using IUE data

IUE has made very successful long term and intense short time-scale monitoring spectroscopic study of NGC 4151, a Seyfert 1 galaxy for over nearly 18 years from its launch in 1978 to 1996. The long-term observations have been useful in understanding the complex relation between UV continuum and emission line variability Seyfert galaxies. In this paper, we present the results of our studies on the short-timescale intense monitoring campaign of NGC 4151 undertaken during December 1–15, 1993. A most intense monitoring observation of NGC 4151 was carried out by IUE in 1993, when the source was at its historical high flux state with a shortest interval of 70 min between two successive observations. We present our results on emission line and continuum variability amplitudes characterized by \(F_{\mathrm{var}}\) method. We found highest variability of nearly 8.3% at 1325 \(\AA \) continuum with a smallest amplitude of 4% at 2725 \(\AA \). The relative variability amplitudes (\(R_\mathrm{max}\)) have been found to be 1.372, 1.319, 1.302 and 1.182 at 1325, 1475, 1655 and 2725 \(\AA \) continuum respectively. The continuum and emission line variability characteristics obtained in the present analysis are in very good agreement with the results obtained by Edelson et al. (1996) and Crenshaw et al. (1996) from the analysis of the same observational spectral data. The large amplitude rapid variability characteristics obtained in our study have been attributed to the continuum reprocessing of X-rays absorbed by the material in the accretion disk as proposed by Shakura and Sunyaev (1973). The continuum and emission light curves have shown four distinct high amplitude events of flux maxima during the intense monitoring campaign of 15 days, providing a good limit on the amplitude of UV variability and the BLR size in low luminosity Seyfert galaxies and are useful for constraining the continuum emission models. The decreasing \(F_{\mathrm{var}}\) amplitude of UV continuum with respect to increasing wavelength obtained in the present study and consistent with similar observations by Edelson et al. (1996) and Crenshaw et al. (1996) is a significant result of the intense monitoring observations.     

The Na <Emphasis Type="SmallCaps">i</Emphasis> and Sr <Emphasis Type="SmallCaps">ii</Emphasis> Resonance Lines in Solar Prominences

We estimate the electron density, \(n_{\mathrm{e}}\), and its spatial variation in quiescent prominences from the observed emission ratio of the resonance lines Na?i?5890 Å (D₂) and Sr?ii?4078 Å. For a bright prominence (\(\tau_{\alpha}\approx25\)) we obtain a mean \(n_{\mathrm{e}}\approx2\times10^{10}~\mbox{cm}^{-3}\); for a faint one (\(\tau _{\alpha }\approx4\)) \(n_{\mathrm{e}}\approx4\times10^{10}~\mbox{cm}^{-3}\) on two consecutive days with moderate internal fluctuation and no systematic variation with height above the solar limb. The thermal and non-thermal contributions to the line broadening, \(T_{\mathrm{kin}}\) and \(V_{\mathrm{nth}}\), required to deduce \(n_{\mathrm{e}}\) from the emission ratio Na?i/Sr?ii cannot be unambiguously determined from observed widths of lines from atoms of different mass. The reduced widths, \(\Delta\lambda_{\mathrm{D}}/\lambda_{0}\), of Sr?ii?4078 Å show an excess over those from Na?D₂ and \(\mbox{H}\delta\,4101\) Å, assuming the same \(T_{\mathrm{kin}}\) and \(V_{\mathrm{nth}}\). We attribute this excess broadening to higher non-thermal broadening induced by interaction of ions with the prominence magnetic field. This is suggested by the finding of higher macro-shifts of Sr?ii?4078 Å as compared to those from Na?D₂.     

Preface

We revisit the relegation algorithm by Deprit et al. (Celest. Mech. Dyn. Astron. 79:157–182, 2001) in the light of the rigorous Nekhoroshev’s like theory. This relatively recent algorithm is nowadays widely used for implementing closed form analytic perturbation theories, as it generalises the classical Birkhoff normalisation algorithm. The algorithm, here briefly explained by means of Lie transformations, has been so far introduced and used in a formal way, i.e. without providing any rigorous convergence or asymptotic estimates. The overall aim of this paper is to find such quantitative estimates and to show how the results about stability over exponentially long times can be recovered in a simple and effective way, at least in the non-resonant case.     

Relativistic satellite orbits: central body with higher zonal harmonics

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \(<\!{\dot{\Omega }}\!>\) and argument of the pericenter \(<\!{\dot{\omega }}\!>\). Finally orders of magnitude are given and discussed.     

VLA Measurements of Faraday Rotation through Coronal Mass Ejections

Coronal mass ejections (CMEs) are large-scale eruptions of plasma from the Sun, which play an important role in space weather. Faraday rotation is the rotation of the plane of polarization that results when a linearly polarized signal passes through a magnetized plasma such as a CME. Faraday rotation is proportional to the path integral through the plasma of the electron density and the line-of-sight component of the magnetic field. Faraday-rotation observations of a source near the Sun can provide information on the plasma structure of a CME shortly after launch. We report on simultaneous white-light and radio observations made of three CMEs in August 2012. We made sensitive Very Large Array (VLA) full-polarization observations using 1?–?2 GHz frequencies of a constellation of radio sources through the solar corona at heliocentric distances that ranged from 6?–?\(15~\mathrm{R}_{\odot}\). Two sources (0842+1835 and 0900+1832) were occulted by a single CME, and one source (0843+1547) was occulted by two CMEs. In addition to our radioastronomical observations, which represent one of the first active hunts for CME Faraday rotation since Bird et al. (Solar Phys., 98, 341, 1985) and the first active hunt using the VLA, we obtained white-light coronagraph images from the Large Angle and Spectrometric Coronagraph (LASCO) C3 instrument to determine the Thomson-scattering brightness [\(\mathrm{B}_{\mathrm{T}}\)], providing a means to independently estimate the plasma density and determine its contribution to the observed Faraday rotation. A constant-density force-free flux rope embedded in the background corona was used to model the effects of the CMEs on \(\mathrm{B}_{\mathrm{T}}\) and Faraday rotation. The plasma densities (\(6\,\mbox{--}\,22\times10^{3}~\mbox{cm}^{-3}\)) and axial magnetic-field strengths (2?–?12 mG) inferred from our models are consistent with the modeling work of Liu et al. (Astrophys. J., 665, 1439, 2007) and Jensen and Russell (Geophys. Res. Lett., 35, L02103, 2008), as well as previous CME Faraday-rotation observations by Bird et al. (1985).