A Semi-empirical Approach to the Dynamic Coupling of CMEs and Solar Wind

Coronal mass ejections (CMEs) are one of the most relevant phenomena for space weather. Moreover, CMEs can negatively affect essential services and facilities. Therefore, to protect society, we require well-grounded knowledge of the physics that governs the propagation of CMEs from near the Sun to the orbit of Earth. In this work, we deduce expressions to approximate the main forces that affect the dynamic coupling between CMEs and the surrounding solar wind. Therefore, we explore the CME–solar wind dynamic coupling from a magnetohydrodynamic perspective, which, combined with a few reasonable assumptions, allows us to obtain expressions for the thermal and magnetic pressure forces, viscous and dynamic drag, and gravity. We simultaneously use our expressions to compute the trajectories of 34 Earth-directed CMEs. Our results, which are compared with in situ data, show significant quantitative consistency; our synthetic transits closely mimic their in situ observed counterparts. We conclude from our results that magnetic, thermal, and dynamic drag significantly surpass the other forces such as dynamic agents of CMEs in the interplanetary medium. In addition, we find that the initial relative speed of CMEs and solar wind is a determinant factor for the dynamic behavior of CMEs. In other words, subsonic CMEs are initially mostly affected by magnetic and thermal pressure forces, whereas super-magnetosonic CMEs are initially governed by inertial drag.


Introduction
Coronal mass ejections (CMEs) are expulsions of magnetized plasma from the solar corona that propagate through the interplanetary (IP) medium with a broad range of speeds ranging from 200 km s −1 up to 3000 km s −1 and masses of around 10 13 kg (Vourlidas et al. 2002(Vourlidas et al. , 2010. CMEs are the main causes of space weather effects on Earth, potentially compromising telecommunications, energy distribution, geopositioning, and other related services and facilities (Echer et al. 2005;Goodman 2005; Kamide & Chian 2007;Moldwin 2008;Schrijver 2015). Therefore, it is necessary to compute accurate forecasting of the properties of CMEs, in particular their arrival time into Earth's vicinity, to better protect our society from these adverse effects. This forecasting is only possible through well-grounded knowledge about the physics that govern CME trajectories from near the Sun to Earth's orbit.
The solar corona can accelerate CMEs through a variety of magnetic-based mechanisms, but they tend to decay suddenly with distance or time (Chen & Garren 1993;Chen & Krall 2003;Vršnak et al. 2004;Chen et al. 2006;Schrijver & Siscoe 2013). Hence, beyond a few solar radii, the effects (acceleration) of those mechanisms that set the initial speed of CMEs become negligible, leaving the interaction with the ambient solar wind as the dominant dynamic agent acting on the trajectories of CMEs (Zhang & Dere 2006;Bein et al. 2011Bein et al. , 2012. Although an approximation, it is physically reasonable that the average acceleration of CMEs can be related to the proper speed of CMEs and the ambient solar wind's expansion speed (Gopalswamy et al. 2000(Gopalswamy et al. , 2001. Such a relationship has been mainly interpreted as a drag force described by a viscosity-like mechanism (Vršnak & Gopalswamy 2002;Cargill 2004), by ram pressure (inertial drag) (Corona-Romero & Gonzalez-Esparza 2011;Corona-Romero et al. 2013), or even by mass accumulation (snow-plug) (Tappin 2006).
Most of the analytical models of the propagation of interplanetary CMEs (ICMEs) that rely on the previously mentioned speed-related drag-mechanisms generally depart from already known expressions that are modified to adapt them to the particular conditions of CMEs. Unfortunately, this procedure usually introduces multiple unknown free parameters that cannot be easily estimated using available data. This procedure tends to decrease the models' efficacy and potentially increase its errors.
Although free parameters are common limitations in most models, they generally come from simplifications based upon evidence, educated guesses, or working hypotheses. Nevertheless, their value is severely limited when there are multiple free parameters related to the very mechanism that is being used to perform the analysis or with the phenomenon that is being analyzed. In this case, the unknown free parameters might indicate a lack of physical information or additional relevant yet ignored mechanisms. Thus, it would be desirable for free parameters to be related to external factors or independent processes and not to the same tools used for the analysis. In addition, it is important to highlight that the previously discussed analytical approaches neglected other forces that might potentially affect the trajectories of CMEs, including magnetic and thermal pressures, as well as magnetic tensions, to mention a few. Their omission can be detrimental to the physical description and interpretation of the phenomenon.
From a magnetohydrodynamic (MHD) perspective, a CME can be seen as an obstacle that propagates within a plasma (the solar wind). This MHD interaction between CMEs and solar wind would include additional dynamic processes (forces), besides viscous and dynamic drags, which simultaneously affect CMEs along their trajectories. To our knowledge, however, no significant attempts have addressed the CMEsolar wind dynamic coupling analytically as an MHD interaction. This forms the primary motivation for the current study.
In Section 2, we explore the forces that arise from the dynamic coupling between the CME and solar wind from an MHD perspective. First, we examine the effect of MHDs on CMEs that are propagating while immersed within the solar wind under a set of necessary simplifications. This effort will lead us to deduce a set of simple expressions that approximate the dynamic agents that, in principle, affect the propagation of CMEs within the inner heliosphere. Subsequently, in Section 3, we explore the capability of our deduced expressions to reproduce the IP propagation of CMEs by using them to compute the trajectories of 34 Earth-directed CMEs. Finally, we present our results and discuss the most relevant conclusions in Section 4.

MHD Dynamic Coupling
As noted earlier, before a CME propagates through the IP medium, the magnetic-based mechanisms that expelled it from the solar neighborhood accelerate it above the Sun's escape speed (∼200 km s −1 ) and also set the speed that launches the CME into the heliosphere. Once a CME begins to propagate through the solar wind, it must be affected by the surrounding solar wind, and possibly, to an ever-decreasing extent, by the forces that led to its expulsion in the first place. This condition suggests that there should be a critical distance (d c ) beyond which solar wind effects become the dominant factor. Thus, in this study, we will assume that, beyond some distance, d c , the acceleration on the CME center of mass (a) can be wellapproximated by where the right-hand side of Equation (1) lists the forces caused by inertial and viscous drag, thermal and magnetic pressure gradients, magnetic tension, and gravity, respectively (see, e.g., Chen 1989;Chen & Krall 2003;Vrsnak et al. 2003;Vršnak et al. 2010, and references there in). On the left-hand side of the Equation (1), M is the CME's total mass. Hence, to determine the trajectories of CMEs in the IP, we need estimates for all the forces listed in Equation (1). We apply the following assumptions to deduce these estimations: (1) The interaction between the CME and solar wind obeys the MHD approximation.
(2) The solar wind is a stationary polytropic and perfectly conductive plasma that radially expands with a Parker-like magnetic field. (3) We approximate the CME geometry as a curved cylinder of circular cross section with a radius that follows a self-similar growth. (4) The CME's mass is constant and uniformly distributed within the CME. (5) The CME's magnetic field is a force-free flux-rope. (6) We neglect the effects of magnetic reconnection between the magnetic fields within the CME and the solar wind. Additionally, but not necessarily, (7) we assume CMEs propagate toward the Earth and near the ecliptic to avoid projection effects in our analysis.
As an example, in panels (a) and (b) of Figure 1, we show a schematic of the so-called "croissant" geometry for CMEs , where the shaded region represents the geometry we used in this work. Panels (c) and (d) of Figure 1 show the upper and frontal views of this shaded region. Figure 1 also illustrates the main variables used for this work, with ñ and R being the mass center position and the radius of the CME, respectively. It is important to remark that the geometry we assume for CMEs is generally equivalent to the well-known "magnetic clouds," also called flux-rope CMEs. Thus, our approach is more suitable for this subgroup of more coherent and organized events; nevertheless, we can adapt or approximate our expressions to almost any required simple geometry.

Simplified Forces
We deduce the forces that arise from the dynamic coupling between the CMEs and the solar wind by integrating the MHD stresses over the CME surface and then applying our assumptions to simplify the resulting expressions. This procedure allows us to derive approximations for all the forces listed on the right side of Equation (1). We show the derivation of those approximations in Appendix A. In the following paragraphs, we show the resulting expressions for the radial components solely because they primarily define the travel times and arrival speeds of CMEs within the Earth's environment.
The first force is the inertial drag, which arises from the resistance of the solar wind's inertia to the CME propagation. This force tends to equalize the relative speed between the solar wind and the CME, accelerating (decelerating) CMEs slower (faster) in relation to the surrounding solar wind. The radial component of this force can be expressed by where m p , n, and u are the proton mass, proton density, and expansion speed of the solar wind, respectively. In addition, the subscripts indicate values relative to ahead of (A, at ñ + R) or behind (B, at ñ − R) the CME. Additionally, v r is the radial component of the CME mass-center velocity,  R the CME expansion speed, and S r the effective radial-oriented surface of the CME. In Appendix B, we present expressions of n, u, S r , and  R as functions of heliocentric distance. The next force we consider is viscous drag, driven by the dissipation of kinetic energy through viscosity. Although the solar wind is a collisionless plasma, multiple proposed mechanisms might produce viscous-like effects, including wave-particle interactions (Williams 1995;Verma 1996), tangled magnetic fields (Subramanian et al. 1996), Landau damping, Bohm diffusion, and Coulomb collisions (Borovsky & Gary 2009). We further note that the concept of the solar wind viscosity hypothesis has been used to explain phenomena observed in the ionospheres of Mars and Venus (see, e.g., Pérez-de-Tejada 2008;Pérez-de-Tejada et al. 2009. Thus, for completeness, we assume that the solar wind has an effective dynamic viscosity (η) that produces a radial drag force approximated by is the Reynolds number, L and  L are the length and longitudinal expansion speed of the CME, respectively, and S f and S θ are the upper (lower) and right (left) effective surfaces of the CME (see Figure 1). We highlight that Equation (3) includes the viscous effect due to the propagation of CMEs, and also the effects of the radial and transverse expansions. It is important to remark that Equation (3) is valid for nonturbulent regimes (R  10 4 ). We neglected the turbulent regime because we could not detect any signature of turbulent wake behind our case studies. As noted earlier, further details on the derivation of Equation (3) can be found in Appendix A.
Next, we consider the forces caused by thermal (Equation (4)) and magnetic (Equation (5)) pressure gradients. Ultimately, these forces derive from the pressures acting on the surfaces of CMEs. The radial component of the thermal force can be approximated by where k B is the Boltzmann's constant and T is the temperature of solar wind protons. The radial force due to magnetic pressure is expressed by where μ 0 represents the magnetic constant and B is the magnitude of the magnetic field.
In the case of the force caused by magnetic tension, we infer that the radial component tends to vanish because the IP magnetic force must be parallel near the CME boundaries: In other words, the geometric configuration produces a negligible inner product between the IP magnetic field near the CME and the CME's surface vector, making the magnetic tension force effectively vanish. We note, however, that this scenario might not be valid when there is magnetic reconnection between CMEs and the IP magnetic field or when plasma resistivity grows larger near the CME-solar wind boundary, allowing magnetic diffusion to occur. However, these two scenarios contradict our assumptions. Finally, the radial component of the gravity force acting on the CME can be approximated by where G and M e are Newton's constant and the solar mass, respectively. The expression of gravity force reflects the fact that CMEs are obstacles immersed inside a fluid (solar wind). Hence, the displaced solar wind's mass must be considered a buoyancy effect. In panels (c) and (d), we approximate the CME material through a cylinder (shaded region) seen from above and laterally, respectively. In the panels, we present the locations of the mass center (ñ), and radius (R) of the CME, its semi-angular (θ) and longitudinal (L) widths, as well as the effective frontal (S r ), latitudinal (S f ), and longitudinal (S θ ) surfaces. We also present the position of the Sun (e). Moreover, the panels present the coordinate system used throughout this work.

Analysis and Results
We explore our approach's capability to approximate the propagation of CMEs by computing the trajectory of 34 CME events. Table 1 shows the events selected as case studies. We emphasize that our criteria for an event's selection rely on the assumptions listed in Section (2). All events were Earth-directed CMEs registered in in situ measurements as they passed 1 au. In general terms, they propagated almost in isolation through the IP medium, and the in situ measurements showed a relatively stable solar wind a few hours before the CMEs transited through the orbit of the Earth; Table 2 shows the input values associated with our study cases. Additionally, we attempted to only select events that propagated in the absence of structures like stream-interacting regions or co-rotating interacting regions that could significantly violate our key assumptions. All of our case studies can be found in the Richardson and Cane (Richardson & Cane 2010), LASCO CME (Yashiro et al. 2004;Gopalswamy et al. 2009), and CACTus (Robbrecht & Berghmans 2004;Robbrecht et al. 2009) catalogs.
To accomplish our analysis, we integrated the speed and position, i.e., computed the trajectories of our case study events through Equation (1). Our computations required initial and boundary conditions inferred from the data associated with each event. We used data from coronagraph images and near-Earth solar wind in situ spacecraft to set up the initial conditions and the ambient solar wind through which the CMEs propagated, respectively. In the following paragraphs,  Notes. From left to right: The event number, event's detection date and hour, initial data from coronagraph images, in situ data for the CME transit through the Earth's orbit, results from the optimization process, and the errors related to our computed trajectories. a Date [YYYYMMDD] and time [hhmm] as reported in SOHO/LASCO CME Catalog. b Data taken from LASCO CME and CACTus Catalogs. (Yashiro et al. 2004;Gopalswamy et al. 2009;Robbrecht et al. 2009). c Travel times from near the Sun to Earth's orbit of the leading and trailing edges of CMEs, and the associated arrival and departure speeds (Richardson & Cane 2011a, 2011b. d Constant governing the CME self-similar expansion, critical distance, and the constant to determine the CME radius length. e Proportional errors associated to our calculated values of TT l , TT t , AS, and DS. The total proportional error (ε) is defined as TT TT AS DS l l we focus on our analysis and results. The details of the computations are provided in Appendix B. We assume that our computed trajectories can be decomposed into two stages: the first where the forces caused by the interaction with solar wind are negligible (turned off), followed by a stage where the MHD forces govern the interplanetary propagation of CMEs. During the first stage, we kept the leading speed of CMEs constant, whereas in the second stage the CME speed is affected by the forces produced by the dynamic coupling between the CME and the ambient solar wind. Finally, we define a critical distance, d c , as the heliocentric distance at which the MHD forces "turn on" and begin to affect the propagation of the ICME.
In our approach, d c aims to approximate the distance at which the MHD effects take over the CME dynamics, either because the mechanisms that expel the CMEs from the solar corona become exhausted by the increase in the heliocentric distance, or because those mechanisms are overwhelmed by the effects of CME-solar wind dynamic coupling. Because of the uniqueness of each CME, the value of d c may vary from one event to another. Thus, for that reason, d c should be considered to be an unknown free parameter in the analysis.
We also require two additional free parameters, one related to the CME radius (R) and the other associated with its semiangular width (θ) (see Figure 1). In the case of θ, we use a constant average value of 30 • (Vourlidas et al. 2002(Vourlidas et al. , 2010, while we express the value of R by the self-similar relation of Gulisano et al. (2010):  Notes. From left to right: The event number, in situ solar wind conditions, in situ data related to the CME transit through the Earth's orbit, and complementary data related with the associated flare and active region (AR), when available. a In situ values for expansion speed, proton density, alpha particle proportion, proton temperature, and IP magnetic field available at OMNIweb PLUS (Papitashvili & King 2020a, 2020b. b Estimated value of the CME mass, and registered values of AS and DS according CME catalogs (Yashiro et al. 2004;Gopalswamy et al. 2009;Robbrecht et al. 2009;Richardson & Cane 2011a, 2011b and in situ signatures (Zurbuchen & Richardson 2006;Wimmer-Schweingruber et al. 2006). c Longitude and latitude of the AR location on the solar disk, class of the associated solar flare, and total angle of the AR (Gallagher et al. 2002). N/A refers to "not available." were d ⊕ is an astronomical unit. We also added the dimensionless parameter k to distinguish longer (k > 1) or shorter (0 < k < 1) CME radii than the average (k ∼ 1). The value of the parameter k is given by the positive root of is the distance the CME went through during its transit to the orbit of the Earth.
In Equation (8), we also have the parameter ò, which controls the self-similar growth of R. The value of ò depends on the solar wind's effects on the CME expansion along the IP medium, where a value of 0.45 (0.89) refers to an expansion highly (barely) perturbed by the solar wind (see, e.g., Gulisano et al. 2010). In this work, we assume ò as the third free parameter, and its value would also be unknown.
Since our approach has unknown free parameters, to compute the trajectories of our study cases, we had to perform an optimization process to find the values of d c and ò that minimized the error (ε l ) for the travel time of the leading edge (TT l ). We defined the error associated with the travel time of the leading edge as where TT l insitu (TT l computed ) is the in situ registered (computed) travel time of the CME's leading edge to the Earth's orbit. Similarly, we can define the errors ε t , ε A , and ε D related to the transit time of the trailing edges (TT t ), arrival speeds (AS), and departure speeds (DS), respectively. When necessary, during the optimization, we used ε t and the total proportional error (ε; see Table 1) as the second and third criteria for optimization, respectively. In our optimization process, we used travel times instead of other parameters, because they are most relevant for space weather purposes; in addition, travel times are less affected by geometrical (projection) effects, because they are scalar quantities.
Our optimization's results are shown in Table 1, where we can appreciate the corresponding proportional errors in columns 12-16. We can verify that our optimization resulted in a median for ε l of 0.0%(0.0 hr) with an associated standard deviation of 0.6%(0.3 hr). Furthermore, the values of ε t , ε A , and ε D were around 0% with standard deviations of 4.5%, 11.3%, and 10.7%, respectively. Moreover, 85% of the study cases had a total absolute error lesser than 20%. Complementarily, the three panels of Figure 2 shows the frequency distribution of ε A , ε D , and ε t . In panel (a), we note that the value of errors related with AS showed a tendency to be distributed around 0 km s −1 , i.e., the computed AS are systematically consistent with their in situ registered counterpart, with differences of ±80 km s −1 , approximately. In the case of ε D (panel (b)), we note a slight tendency for DS to leave the Earth's orbit faster because computed departure speeds are faster than their in situ registered counterparts. Nevertheless, the values of the median and standard deviation of ε D are short compared with the measured DSs. Finally, in panel (c), we note that the error distribution associated with computed TT t is almost centered around 0 h, with a deviation of ±4 h. Hence, the errors indicate that the arrivals and departures derived from our computed trajectories are quantitatively consistent with their in situ registered available data.
The consistency between computed trajectories and the data associated with the study cases can also be appreciated in the in situ transits of the events. In the upper panels (a) and (b) of Figures 3, 4, and 5, we show comparisons between the in situ transits of CMEs and their computed counterparts. On the one hand, in panels (a), we plot the in situ speed (gray line) of the solar wind near Earth's orbit and the computed transit speed (solid red line). In panels (b), we plot the in situ measured proton density (gray line) and our resulting CME proton density (solid red line). In these three figures, we note that our computed transits of speed (panels (a)) and proton density (panels (b)) closely mimic in tendency and value their in situ registered counterparts, which we delimited by dashed vertical black lines in panels (a) and (b). In combination, the quantitative agreement with CME arrivals and departures and the consistency between in situ registered and computed transits suggest that our computed trajectories may also be consistent with those our study cases should have had.
We proceed to present the resulting trajectories. However, before presenting them, we need to highlight that we could identify three well-differentiated classes of computed trajectories. Each class is defined by the behavior of the total force (acceleration) that dominated the event's propagation. First, we have those with positive total force, and next, we have those that showed negative total force. Finally, an intermediate case that began with a positive force turned negative at some point The computed trajectory of Event 27, as seen in panels (c) and (d) of Figure 3, is an example of the positive total force class. In panel (c), we note that it began its propagation with an initial leading speed of ∼449 km s −1 (open red square), which stayed constant up to d c (t c ), after which the MHD effects activate. In panel (e), we appreciate the different accelerations (forces) that affect the CME beyond d c , with the total acceleration (solid thick black line) being the result of all those combined effects. In this case, we note that the total Computed accelerations (e) due to all dynamic effects (colored profiles) acting on the CME center of mass, and the solid black line is the total acceleration (total force per unit of mass). In panels (a) and (b), we use one-minute data from OMNIweb service to plot the in situ data profiles. In panels  Computed accelerations (e) due to all dynamic effects (colored profiles) acting on the CME center of mass, and the solid black line is the total acceleration (total force per unit of mass). In panels (a) and (b), we use one-minute data from OMNIweb service to plot the in situ data profiles. In panels acceleration starts positive and rapidly decays with time. In panel (d) of the figure, we can appreciate this acceleration's effect since the mass center's speed (dotted-dashed black line) accelerates for a brief period, after which it seems to achieve an almost constant value.
In panel (d) of Figure 3, we note that the computed trajectory of the CME leading edge (solid black line) consistently follows the coronograph data (open red squares). We also appreciate that eventually the leading and trailing (dashed black line) edges of the CME reach d ⊕ (black circles), an arrival that is in agreement with the in situ registered arrival (open blue diamond). It is important to remark that, in this case, despite the similarities between the initial speed of the event and the initial speed of the solar wind, the value of AS was larger than both speeds, contradicting the interpretation of inertial and viscous drags, which are supposed to equalize the solar wind and CME speeds. Nevertheless, in panel (e) of that figure, we appreciate that this unexpectedly faster AS is due to the effect of the already explained positive acceleration mainly caused by the combined effects of magnetic (solid blue line) and thermal (solid cyan line) pressures. It is important to point out that these relevant effects for the dynamics of slow CMEs are commonly neglected in the analytic approaches of the dynamics of CMEs in the IP medium.
On the other hand, Event 17 ( Figure 4) is an example from the negative total force set. Similarly to Event 27, the computed transits for these study cases are consistent with their in situ counterparts. This event, however, significantly differs from the last one in the speed at which it was initially detected. The calculated speed of Event 17 from coronagraph images was ∼1216 km s −1 , which is highly super-magnetosonic when compared with the solar wind speed (∼300 km s −1 ). Furthermore, such a significant difference in speeds led the inertial drag to become the dominant dynamic agent for this event. In the figure's panel (e), we observe that the total acceleration is negative and mainly defined by the effects of the inertial drag (solid red line). The events dominated by negative acceleration (force) tend to be super-magnetosonic from their initial detection, a condition that leads the inertial drag to be the main dynamic agent acting on the events of this set. Event 4 is an example of an intermediate class event (Figure 5). The initial speed of Event 4 was ∼726 km s −1 , which is larger than the solar wind speed (∼530 km s −1 ); however, such a difference is not enough to provoke inertial drag to overwhelm the effects of magnetic and thermal pressures near d c . In Figure 5 (e), we appreciate that the total acceleration begins positive, driven mainly by the thermal (cyan) and magnetic (blue) pressures in almost equal amounts. This period provokes a significant increase in the CME's mass center speed that reaches a maximum value near 20 h as shown in panel (c). We also note that, during the period between t c (temporal counterpart of d c ) and 20 hr, the magnetic and thermal pressures rapidly decay; in contrast, the inertial drag (solid red line) magnitude shows a slower decaying rate. This behavior eventually causes, at ∼25 hr, the total acceleration to turn negative due to the dominance of inertial drag over the thermal and magnetic pressures. Although Event 4 is still decelerated when it arrives at d ⊕ , the initial positive acceleration provoked an AS (∼795 km s −1 ) similar in magnitude to the initial one. Thus, although this event was initially faster than the solar wind, the thermal and magnetic effects were determinants of its arrival conditions.

Discussion and Concluding Remarks
We have developed expressions to approximate the main forces that, in principle, govern the dynamic coupling between CMEs and the surrounding solar wind. In order to deduce those Computed accelerations (e) due to all dynamic effects (colored profiles) acting on the CME center of mass, and the solid black line is the total acceleration (total force per unit of mass). In panels (a) and (b), we use one-minute data from the OMNIweb service to plot the in situ data profiles, and we use 1 hr average data to fill the gaps. In panels expressions, we explored the dynamic effects of solar wind on CMEs propagating through it from an MHD perspective. Subsequently, we simplified those dynamic effects by applying some assumptions to obtain easy-to-use expressions. Our approach's dynamics effect includes ram, thermal and magnetic pressure, viscous drag, and gravity forces. We also explored magnetic tension and induced Lorentz force; however, the effects of those forces vanished because of our assumptions and the physical nature of such forces. Nevertheless, a different geometry for CMEs or complex magnetic configurations (nonforce-free flux rope) inside magnetic clouds could not cancel the Lorentz force. Similarly, the effects of magnetic reconnection may also cause the effects of magnetic tensions not to be negligible.
We used our deduced expressions to compute the trajectories of 34 CME events listed in Table 1. We highlight that our computing required three free parameters: d c , ò, and θ. The first (d c ) is related to the reaching distance of the physical mechanisms that expel (accelerate) CMEs out from the solar corona and set the initial conditions for CMEs to propagate through the solar wind. The second one (ò) is a constant that governs the self-similar growth rate of CME's radius, which has a value in principle that is settled by the solar wind's effects over the CME expansion. Finally, θ defines the semi-angular width of CMEs, for which we used the average value of 30°for all our study cases.
To compute the trajectories for our selected events, we performed an optimization process to find the values of our two unknown free parameters (d c and ò) that minimized the error in the travel times of the CME leading edge (TT l ). The median error of the TT l associated with the computed trajectories, which resulted from our optimization, was 0.0%(0.0 h) with a standard deviation of 0.6%(0.3 h). In this work, we used TT l instead of other parameters like speeds, because it is most relevant for space weather purposes; in addition, travel times, as a scalar quantity, are less affected by geometrical (projection) effects. The ranges of values covered by our optimization were 5R e d c 150R e and 0.45 ò 0.89.
The consistency between computed trajectories and the data associated with the study cases can also be appreciated in the in situ transits of the events. The upper panels ((a) and (b)) of Figures 3, 4, and 5 show comparisons between the in situ transits of CMEs and their computed counterparts. In the figures, we note that our computed transits closely mimic their in situ registered counterparts. The low value of errors in CME arrivals/departures and the consistency in approximating in situ transits suggest that our computed trajectories may also be consistent with the ones the study cases may have had.
In Figures 3, 4, and 5, panels (c) and (d) show examples of computed speeds and heliocentric distances for Events 27, 17, and 5, respectively. In the panels (c) of the figures, we note two stages in the speed profiles. The first is characterized by a constant speed of the leading edge of CMEs that departs from its initial condition (open red squares) settled by data from coronagraph images. This stage finishes at the heliocentric distance d c , at which the process that expels the CME from the solar corona is exhausted or is also overwhelmed by the effects of the surrounding solar wind. The subsequent stage in the speed profile is dominated by the effects of the CME-solar wind dynamic coupling.
Each one of the previously mentioned figures shows a different class of event determined by the behavior of the acceleration (as seen in panels (e) of the figures) acting on it: positive (Figure 3), negative (Figure 4), and intermediate ( Figure 5). In the case of Event 27 (Figure 3(c)), we note the CME speed rapidly increases, and it achieves an almost constant value before arriving at Earth's orbit (open blue diamonds). In contrast, Event 17 (Figure 4(c)) suffers a strong deceleration that seems to gradually decay with distance. Surprisingly, Event 5 ( Figure 5(c)) shows a first short period of significant acceleration, followed by a prolonged deceleration.
When we explore the relative speed between the CME initial speeds and solar wind expansion speed (  ℓ u 0 1 -) of Events 27, 17, and 5, we obtain ∼−70 km s −1 , ∼900 km s −1 , and ∼200 km s −1 , respectively. Hence, we observe that the initial relative speed of CMEs may determine the dynamical evolution commented in the paragraph above. In the subsonic case (Event 27, ⪅  ℓ u 0 1 ), our results indicated that magnetic and thermal gradient pressures are the main mechanisms that accelerate the CME away from the Sun. These gradients are permanently directed away from the Sun, because the solar wind's magnetic and thermal pressures decrease with heliocentric distance, the first decaying as ∼r −4 and the latter as ∼r −1 . In addition, such a decrease also provokes a rapid intensity-fall of these gradient pressure forces. In contrast, the super-magnetosonic case (Event 17,   ℓ u 0 1 ) is dominated by the ram pressure. The ram pressure rises from the CME-solar wind relative speeds and is modulated by the solar wind density that falls as ∼r −2 ; however, as long as the relative speed is not null, the ram pressure effects endure. The latter explains the negativity of ram pressure in the super-magnetosonic case, because the solar wind opposes resistance to the CME propagation. In the case of the trans-magnetosonic event (Event 5,  ℓ u 0 1 ), we note an initial behavior similar to a subsonic event, i.e., it is dominated by magnetic and thermal pressures. For this case, however, the ram pressure turns negative beyond d c and decays more slowly than the magnetic and thermal counterparts, which causes the total acceleration to finally turn negative at ∼25 h, which is the behavior we observed for super-magnetosonic events. Hence, in the case of the trans-magnetosonic event, we notice a brief stage of acceleration driven by magnetic and thermal gradients, followed by long-lasting deceleration dominated by ram pressure.
We concluded from our results that magnetic, thermal, and ram pressures significantly surpassed the other forces as dynamic agents for CMEs' propagation in the IP medium. In contrast, the effects of gravity and viscous forces were negligible on our computed trajectories. Nevertheless, we also highlight that another phenomenon might indirectly affect super-magnetosonic events and is missing in our approach: interplanetary shocks associated with CMEs. Interplanetary shocks modify (compress and accelerate) the solar wind in which super-magnetosonic CMEs propagate. Thus, although our expressions for the forces must not change, the presence of shocks would modify the solar wind we used to compute the CME trajectories and also modify the forces' behavior. This condition might tentatively affect our results for super-magnetosonic and trans-magnetosonic events. Figure 6 shows potential evidence for this; in the figure, we appreciate the absolute total error of speeds ( A   D   2 2 e e + ) as a function of the relative speed of CMEs and solar wind as seen at d ⊕ . In the figure, we note a tendency (gray dashed line) for the error to grow with the relative speed. In addition, in the figure, we also differentiate events according to their magnetosonic Mach number (M ms ), revealing that events with larger values of M ms > 2 that are indeed preceded by IP shocks tend to be associated with more significant errors. Thus, this error tendency to grow with the relative speed and magnetosonic Mach number could be a tentative effect of IP shocks, which is missed in our approach. Hence, the effects of IP shocks on the dynamics of trans and super-magnetosonic CMEs should be explored, which is a task that we leave as future work.
Our approach succeeded in quantitatively approximating the trajectories and in situ transits of events with initial speeds between 236 km s −1 and 1384 km s −1 in a diversity of solar wind conditions. This was done by optimizing the two unknown free parameters related to the expansion of CMEs and the range of the mechanisms that expel CMEs from the solar corona. We emphasize that our free parameters are not directly related to our approach. In the case of d c , it refers to an external factor: the reaching distances of the expelling mechanisms of CMEs. Regarding ò, it governs the expansion of CMEs, a physical process different from the one studied in this work.
Perhaps this approach's main limitations are the assumed geometry for CMEs, the isolated propagation requirement, and the stationary condition for solar wind. The first limitation, however, can be easily addressed, and the remaining two might be solved by combining this analytic approach with numerical solar wind simulations as inputs. Additionally, it would be convenient to smooth the turn-on of forces instead of the abrupt initiation we used at d c .
Finally, we remark that our approach proceeds from assumptions that may oversimplify the physical complexity of CMEs and the dynamic interaction CMEs have with the solar wind. Undoubtedly, we should extend our approach to cover more complex and general conditions like contemplating structured solar wind (co-rotating or stream-interaction regions), CMEs with "collision interaction," or non-Sun-Earth-aligned CMEs, to mention a few. Nevertheless, our simplified approach could satisfactorily approximate trajectories and arrivals of CMEs in various conditions; this highlights our approach's significant potential for describing the CME-solar wind dynamic coupling. This research has been supported by the Consejo Nacional de Ciencia y Tecnología (grant nos. CB-254812, LN 315829, Catedras CONACYT 1045, andCONACYT-AEM 2017-01-292684). The program "Investigadores por México" (Catedras CONACYT), project 1045, sponsors the Mexican Space Weather Service (SCIESMEX).
We acknowledge the use of NASA/GSFC's Space Physics Data Facility's OMNIWeb service, and OMNI data. This work used data from the CME LASCO Catalog, this catalog is generated and maintained at the CDAW Data Center by NASA and The Catholic University of America in cooperation with the Naval Research Laboratory. SOHO is a project of international cooperation between ESA and NASA. We also acknowledge the use of CACTus CME catalog.

Appendix A CME-Solar Wind Interaction from an MHD Perspective
We can express the effects of MHD stresses (τ ij ) and the force of gravity (F g i ) on the whole CME through On the left side of Equation (A1), ρ and v i are the mass density and velocity inside the CME volume (Ω), respectively, whereas on the right side of the equation, we integrate the effects of τ ij over the whole CME. This integral transforms into a surface integral acting over the CME surface (∂Ω), while gravitational force acts on the CME's mass center. The left side of Equation (A1) can be manipulated to express the product of the mass-center acceleration (a i cme ) of the CME and its mass: In the first term on the left side of the first row of Equation (A2), we can take out the time derivative from the volume integral operator because there are no CME material fluxes nor solar winds through ∂Ω. Subsequently, we use M to allow us to introduce the mass center's velocity (v). In this step, we assumed M to be constant because CME mass shows a tendency to barely change for heliocentric distances larger than a few solar radii (Colaninno & Vourlidas 2009;Bein et al. 2013). Finally, we obtain the time derivative of v: the CME mass center's acceleration (a i ).
The first term on the right side of Equation (A1) accounts for the effects of the stress tensor that groups the MHD dynamic agents present in the CME-solar wind interaction. Such agents include ram and thermal pressures, viscous effects, and magnetic with v the velocity of the solar wind relative to the CME surface, and ρ the density of solar wind near the CME-solar wind boundary. In the second row of Equation (A4), we approximate the ram pressure effects over the whole CME surface (∂Ω) through the effects over the idealized surfaces S r ±, S θ ±, and S f ±(see Figure 1). The signs (±) indicate the different directions of each surface, i.e.,, front/rear for r, up/ down for f, and right/left for θ. As a first approximation, non-radial (i ≠ r) components of inertial drag vanish or are not significant because of the assumed symmetry on solar wind properties and the opposite directions of the surfaces S θ ± and S f ±. Hence, in a fairly unstructured IP medium, the ram pressure or inertial drag (F in ) is mainly radial (i = r) and would be approximated by Equation (2).
The second term of τ ij concerns the viscous force of the solar wind acting on CMEs: In the first row of the last Equation, v is the solar wind relative velocity near the CME boundaries. In the second row, we approximate the integral over ∂Ω cme via summations on S j and S i , where the ± signs indicate the orientation of the surfaces, as in the case of the inertial drag. Here, we remark that the diagonal of the term related to η is null, which is the reason for the i ≠ j restriction. Finally, in the third row of Equation (A5), ∇ i v j vanishes due to symmetry effects, because the effects of opposite surfaces cancel each other.
From hydrodynamics, we know that viscosity tends to cancel the relative CME-solar wind speed inside the boundary layer, of width R R 1 2 (Landau & Lifshitz 2005b). Hence, the relative speed has an effective gradient inside the boundary layer that we can approximate as Thus, the radial component of the η term reduces to with S f and S θ as the CME surfaces on the f and θ coordinates, respectively. Following a similar procedure, we can estimate ∇ i v i to approximate the term-related effects with second viscosity. We should not neglect those effects, because the solar wind is a compressible fluid and the CME expands radially and longitudinally. Thus, we express: Finally, the radial component of Equation (A5) is given by combining Equations (A6) and (A7) in Equation (3), where we assume for simplicity that η ∼ ζ. The value of η is computed with the hybrid viscosity as defined by Equation (2.13) of Subramanian et al. (1996).
It is important to remark that Equation (3) is solely valid as long the mainly viscous interaction occurs inside the boundary layer (R  10 4 ); otherwise, the viscous drag becomes turbulent. In a turbulent regime, the interactions between the fluid and the obstacle significantly change, which is a condition that invalidates the simplifications used to deduce Equations (A6) and (A7) (Landau & Lifshitz 2005b). We neglect the turbulent regime because, to our knowledge, there are no reports in the literature of a turbulent wake behind CMEs, nor have we detected any in our analyzed events.
The third term on the right side of Equation (A3) represents the effects of the thermal pressure gradient. We can approximate the force produced through its effects by where p is the thermal pressure of solar wind evaluated on the surface S i of the CME. Once again, we neglect the non-radial components because they cancel each other due to symmetry (antisymmetry) of solar wind thermal pressure (CME surfaces), whereas the radial component would be expressed by Equation (4)  In the second row of Equation (A9), the inner product B j ds j vanishes because the IP magnetic field cannot permeate throughout the CME surface because B j is tangent (perpendicular) to the CME surface (normal vector ds j ). Hence, the magnetic tension term vanishes; meanwhile, as previously happened with the thermal pressure force, only the radial component of the magnetic pressure term is of relevance. Finally, we deduce the gravitational force, which for definition, is radial. In order to approximate this force, we must emphasize that CMEs are obstacles immersed inside a fluid (solar wind). Hence, the gravitational force would be a combination of the proper gravitational force acting on the CME and the effects of the solar wind volume displaced by the CME, i.e., a buoyancy force: In the second row of Equation (A10), we approximate the integral over the displaced solar wind volume since ξ/ñ = 1, which is an expression that directly leads to Equation (7).

Appendix B Analysis Details
In this section, we show the expressions for performing our analysis and how we set up the initial and boundary conditions. We begin with the stationary solar wind, with its properties given by the next set of equations: In the last set of equations, r represents an arbitrary heliocentric distance, ω e the solar gyrofrequency at the solar equator, and R e a solar radius. In addition, n 1 , T 1 , u 1 , and B 1 are the density and temperature of protons, speed, and magnetic field magnitude of solar wind measured in situ (at d ⊕ ), respectively. Equation (B1) is the empirical relation by Leblanc et al. (1998), whereas p 1 represents the thermal pressures of electrons and protons, with temperatures and densities that we assume are equal for simplicity. In Equation (B3), γ is the polytropic index that we express through the empirical model by Jacobs & Poedts (2011) In addition, we express the CME mass (M) through the average values of proton density (n p ) and alpha-particle ratio (n a ) measured during the in situ transits of CMEs: where, for simplicity, θ ∼ 30°for all study cases. As mentioned throughout the manuscript, our computed trajectories would be divided into two stages: one for ñ d c and the other for larger heliocentric distances. In the case of the first stage, we force our computed trajectories to be consistent with the coronograph-observed data. Hence, to compute the positions (ñ(t)) and speeds (v(t)) of the CME mass center and radius (R(t)) for a given time (t 0), departing from coronagraph data, we use where R ℓ = R(ℓ), and ℓ 0 and  ℓ 0 are the initial heliocentric distance and speed, respectively, of the CME leading edge as seen in coronagraph images. Once ñ(t) > d c (i.e., t > t c ), we proceed to compute the values of v and ñ by integrating Equation (1) through a fourth-order Runge-Kutta integrator, with the conditions defined at t = t c and r = d c .
Finally, regarding our optimization process, we computed the trajectories of all our study cases by using a variety of values of ò and d c that covered all the possible combinations from a given predefined range of values. The range of values for ò was 0.45 ò 0.89 with increments of 0.055; it is important to remark that such a range is defined in Gulisano et al. (2010). In the case of d c , we used the range of values from 5R e to 150R e divided into 500 steps of equal longitude. For practical reasons, this range begins near the initial position of CMEs and finishes at a heliocentric distance that doubles the largest critical distance (∼75 R e ) reported in a previous work (see Table 2 of Corona-Romero et al. 2013). This procedure led us to compute ∼4500 numerical solutions for each event, from which we searched for the one with the lowest error.