1 Introduction

Prediction of the arrival of interplanetary coronal mass ejections (ICMEs) at 1 AU is one of the primary tasks of the space-weather forecasting, since ICMEs are responsible for major geomagnetic storms (e.g., Koskinen and Huttunen, 2006). Thus, the research on heliospheric dynamics of ICMEs is essential in developing and advancing the forecast methods. The substantial progress in the observational aspect of the research achieved over the past decade is mainly related to the unprecedented observations gathered by the Large Angle Spectroscopic Coronagraph (LASCO; Brueckner et al., 1995) onboard the Solar and Heliospheric Observatory (SoHO), the Solar Mass Ejection Imager (SMEI; Jackson et al., 2004), and especially, the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI; Howard et al., 2008) onboard the Solar Terrestrial Relations Observatory mission (STEREO-A and STEREO-B spacecraft). A significant contribution was provided also by meticulous analysis of radio-spectrographic measurements (e.g., Reiner, Kaiser, and Bougeret, 2007, and references therein) and radio-scintillation measurements (e.g., Manoharan, 2010, and references therein), as well as by in-situ solar-wind measurements from a number of space missions.

The advancement of observational techniques contributed to progress in various forms of modeling the heliospheric propagation of ICMEs, and maybe even more important, provided detailed testing of the related forecast procedures, either from the statistical point of view or in the form of case studies (for the validation of forecast methods see, e.g., Cho et al., 2003; Dryer et al., 2004; Oler, 2004; Owens and Cargill, 2004; McKenna-Lawlor et al., 2006, 2008; Tappin, 2006; Feng et al., 2009; Smith et al., 2009; Byrne et al., 2010; Falkenberg et al., 2010; Maloney and Gallagher, 2010; Taktakishvili et al., 2009; Vršnak et al., 2010; Falkenberg et al., 2011 and references therein). The modeling and forecasting methods can be divided into several classes. On one side, there are purely empirical/statistical methods, or kinematical-empirical methods, based on various relationships between the coronographically measured parameters and the ICME arrival time and/or characteristics of their heliospheric propagation (e.g., Brueckner et al., 1998; Gopalswamy et al., 2000; Vršnak and Gopalswamy, 2002; Manoharan et al., 2004; Michałek et al., 2004; Schwenn et al., 2005; Manoharan and Mujiber Rahman, 2011, and references therein). On the other side, there are numerical MHD-based models of the heliospheric propagation of ICMEs or shocks they are driving (e.g., McKenna-Lawlor et al., 2002; Fry et al., 2003; González-Esparza et al., 2003; Dryer et al., 2004 Manchester et al., 2004; Odstrcil, Riley, and Zhao, 2004; Odstrcil, Pizzo, and Arge, 2005; Smith et al., 2009; Taktakishvili et al., 2009, and references therein).

In between the empirical and numerical methods, there is a class of analytical, MHD- or HD-based, kinematical models of the ICME propagation. Most of them rely on the hypothesis that beyond a certain distance the ICME dynamics becomes governed solely by the interaction of the ICME and the ambient solar wind (Cargill 2004; Owens and Cargill 2004; Vršnak and Žic 2007; Vršnak, Vrbanec, and Čalogović 2008; Borgazzi et al. 2009; Lara and Borgazzi 2009; Vršnak et al. 2010). This assumption is founded on the fact that in the interplanetary space fast ICMEs decelerate, whereas slow ones accelerate, showing a tendency to adjust their velocity to the ambient solar wind (e.g., Gopalswamy et al., 2001; Vršnak et al., 2004; Yashiro et al., 2004; Manoharan, 2006; Vršnak, Vrbanec, and Čalogović, 2008; Morrill et al., 2009; Webb et al., 2009).

In this paper we present a specific version of the drag-based model (hereafter DBM) that provides an analytical solution of the equation of motion, and we analyze its potentials for the real-time space-weather forecasting. The model is based on the equation of motion where the drag acceleration has a quadratic dependence on the ICME relative speed. This is expected to be an appropriate parametrization in the collisionless solar-wind environment, where the drag is caused primarily by the emission of MHD waves (Cargill et al. 1996; Owens and Cargill 2004). The presented version of DBM provides explicit-form solutions for the Sun–Earth ICME transit time and the impact speed (cf. Section 2). This offers very easy handling and straightforward/prompt application in the real-time space-weather forecasting. In Section 3 we present an analysis of the DBM performances, applying the statistical and case-study approach. The advantages and drawbacks of DBM are discussed in Section 4. Finally, in the Appendix we present a public, DBM-based, online forecast tool.

2 The Model

2.1 Model Description

The DBM is based on the assumption that the dynamics of ICMEs is dominated by the MHD “aerodynamic” drag (Cargill et al. 1996; Vršnak 2001a; Owens and Cargill 2004; Cargill 2004; Vršnak et al. 2004, 2010; Vršnak and Žic 2007; Vršnak, Vrbanec, and Čalogović 2008; Borgazzi et al. 2009; Lara and Borgazzi 2009), i.e., that ICMEs which are faster than the ambient solar wind are decelerated, whereas those slower than solar wind are accelerated by the ambient flow (cf., Gopalswamy et al., 2000). In particular, we consider the quadratic form for the drag acceleration (cf., Cargill, 2004 and references therein):

$$ a = -\gamma(v-w)\vert v-w\vert , $$
(1)

where v is the instantaneous ICME speed and w is the ambient solar-wind speed. It is important to note that Equation (1) defines the instantaneous acceleration, i.e., each quantity (a, v, and w) is a function of time, t. The drag parameter γ can be expressed as

$$ \gamma= \frac{c_{\mathrm{d}} A \rho_{\mathrm{w}}}{M+M_{\mathrm{v}}} , $$
(2)

where c d is the dimensionless drag coefficient (Cargill, 2004), A is the ICME cross-sectional area, ρ w is the ambient solar-wind density, and M is the ICME mass. The so called virtual mass, M v, can be expressed approximately as M v∼ρ w V/2, where V is the ICME volume (for details see, e.g., Cargill, 2004 and references therein). Thus, taking into account M=ρV, where ρ is the ICME density, the parameter γ can be expressed also as

$$ \gamma= \frac{c_{\mathrm{d}} A \rho_{\mathrm{w}}}{V(\rho+\frac{\rho_{\mathrm{w}}}{2})} = \frac {c_{\mathrm{d}}}{L (\frac{\rho}{\rho_{\mathrm{w}}} +\frac{1}{2})} , $$
(3)

where L is the thickness of the ICME in the radial direction, and we approximated V∼AL. Equation (3) shows that in the limit ρ w≫ρ the parameter γ does not depend on the solar-wind density (γ=2c d/L). On the other hand, in the regime ρ≫ρ w the virtual mass becomes negligible and one finds γ=c d ρ w/Lρ, also meaning that γ becomes much smaller than in the ρ w≫ρ case.

For the matter of illustration, let us consider the ρ≫ρ w regime and assume that the CME is ten times denser than the ambient solar wind. Taking that at typical coronagraphic distances the CME radial thickness is on the order of one solar radius, say, 106 km, for c d=1 one finds γ∼10−7 km−1.

Generally, the value of γ changes with distance, so it is also dependent on time implicitly. Thus, the equation of motion reads

$$ \frac{\mathrm{d}^2r}{\mathrm{d}t^2} = - \gamma(r) \biggl(\frac{\mathrm{d}r}{\mathrm{d}t} - w(r) \biggr) \biggl \vert \frac{\mathrm{d}r}{\mathrm{d}t} - w(r)\biggr \vert , $$
(4)

where r is the heliospheric distance of the ICME leading edge. In general, Equation (4) has to be solved numerically (Vršnak and Žic 2007).

Equation (4) is fully specified only if the dependencies γ(r) and w(r) are defined. This implies that we have to specify A(r), ρ w(r), M(r), and c d(r). In the simplest form of the model (which is used hereinafter) we assume that at sufficiently large distances (say, r>20r ⊙, where r ⊙ is the solar radius) the following approximations are valid: A∝r 2, ρ w∝1/r 2, M=const., and c d=const. (for details, see Cargill, 2004). Furthermore, we assume ρ≫ρ w, i.e., we consider the case when the effect of the virtual mass M v is negligible. Under such assumptions we get γ(r)=const. (note that Vršnak and Gopalswamy, 2002 considered also the case γ(r)≠const.). Moreover, by taking into account the equation of continuity for the isotropic flow, the condition ρ w∝1/r 2 implies also w=const. (for details see, Vršnak et al., 2004; Vršnak and Žic, 2007). In such a case, Equation (4) can be solved analytically (Vršnak et al. 2004). Note that the approximation w=const. for r>20r ⊙ is consistent with the empirical solar-wind speed models proposed by Sheeley et al. (1997) and Leblanc, Dulk, and Bougeret (1998). The approximation A∝r 2 is consistent with the fact that at typical coronagraphic distances the CME angular width remains constant, i.e., that CME expands in a “self-similar” manner.

For the matter of illustration, let us consider the CME sample employed by Vršnak, Vrbanec, and Čalogović (2008), where masses were in the range M∼1012 – 1013 kg. Employing a typical limb-CME angular width of ϕ=60∘∼1 rad (Vršnak et al. 2007), and applying the cone model (Zhao, Plunkett, and Liu 2002), one finds that at r=20r ⊙ the cross-sectional area is A=(rϕ/2)2 π∼1020 m2. Using c d=1 and a typical solar-wind density of n∼109 m−3 for this distance range (Leblanc, Dulk, and Bougeret 1998), we find γ∼0.2 – 2×10−7 km−1, consistent with the previous estimate based on Equation (3).

The analytical solutions of Equation (4), with the approximation γ(r)=const. and w(r)=const., read

$$ v(t)=\frac{v_{0}-w}{1 \pm\gamma(v_{0}-w) t}+w , $$
(5)

and

$$ r(t)= \pm\frac{1}{\gamma}\ln\bigl[1 \pm\gamma(v_{0}-w)t \bigr]+wt+r_{0} , $$
(6)

where ± depends on deceleration/acceleration regime, i.e., it is plus for v 0>w, and minus for v 0<w.

Thus, in the simplest option of DBM, the solution of Equation (4) can be presented in the form r(t), v(t), or v(r) explicitly. From this we can find the time T needed for an ICME to travel from a given initial (t=0) radial distance, r 0, to 1 AU, for a given initial speed (“take-off speed”), \(v_{0} = v_{t=0} \equiv v_{r=r_{0}}\). In addition, the function v(r) provides the “impact speed”, v 1, i.e., the ICME speed at 1 AU.

We have compared the analytical solutions for γ=const. and w=const. with the numerical outcome for γ(r)≠const. and w(r)≠const., where we employed the interplanetary density model proposed by Leblanc, Dulk, and Bougeret (1998) (for details, see Vršnak et al., 2010). The comparison shows that the difference becomes very small beyond r∼20r ⊙; the deviations in the calculated transit times are generally within 1 h, which is much smaller than the uncertainties introduced by the limited measurement accuracy of the input parameters.

In Figure 1 several examples of the ICME kinematics, calculated employing the analytical solutions given by Equations (5) and (6), are presented. The kinematics of fast ICMEs is illustrated for various values of γ by taking the take-off speed of v 0=1000 km s−1 (full lines), whereas slow ICMEs are represented using v 0=200 km s−1 (dashed lines). Furthermore, we consider propagation in a slow (w=400 km s−1) and fast (w=600 km s−1; gray lines) solar-wind environment.

Figure 1
figure 1

Examples of ICME kinematics based on DBM; the initial heliocentric distance is set to r=20r ⊙ (R=20). (a) Heliocentric distance versus time; (b) ICME speed versus time; (c) ICME speed versus distance; (d) ICME acceleration versus distance. The applied parameters are written in the legends in (a) and (d), where v 0 and w are expressed in km s−1 and Γ=γ×107 km−1 (i.e., γ=Γ×10−7 km−1).

Full size image

Figure 1a shows the ICME heliocentric distance (R≡r/r ⊙) versus time. The ICME Sun–Earth transit time is defined by the intercept of the curves and the upper graph-boundary set at R=214. The ICME speed presented as a function of time is displayed in Figure 1b. The kinematical curves end at R=214, so the x-coordinate of their endpoint defines the ICME speed at 1 AU. The graph showing the ICME speed versus distance is presented in Figure 1c. The 1 AU speed is defined by the intercept of the curves and the right graph-boundary, set at R=214. Finally, in Figure 1d we show the ICME acceleration presented as a function of heliocentric distance.

Figures 1b and c show clearly the tendency of ICMEs to adjust to the solar-wind speed. The adjustment is faster under high-γ conditions, representing “light” ICMEs in a relatively dense solar wind. The transit times range from ∼ 40 to ∼ 110 h, grouping around ∼ 80 h, consistent with the so called “Brueckner 80 h rule” (Brueckner et al. 1998). Note that under the high-γ conditions an initially slow ICME, launched into fast solar wind, can have shorter transit time than an initially fast ICME launched into slow solar wind. Figure 1 also implies that the shortest transit times are achieved if a very fast ICME is launched into fast solar wind under low-γ conditions (massive/narrow ICME in low-density high-speed solar-wind stream). Finally, it should be emphasized that in the high-γ case the proposed simplest form of the model (γ=const.) might not be fully appropriate, since in this regime the virtual mass might not be negligible, implying that γ changes with the distance.

Figure 1d reveals that most of acceleration/deceleration occurs close to the Sun, say, within R∼20 – 40). The calculated accelerations are compatible with typical accelerations measured in the SoHO/LASCO-C3 field-of-view (Vršnak 2001a; Vršnak et al. 2004; Vršnak, Vrbanec, and Čalogović 2008). The mean accelerations over the distance range R=20 – 214 are found to be within ± 2 m s−2 for ICMEs whose take-off speed is relatively close to the solar-wind speed. The deceleration goes up to − 10 m s−2 for very fast ICMEs in slow solar wind under high-γ conditions. Such values are consistent with measurements presented by, e.g., Gopalswamy et al. (2000, 2001) and Michałek et al. (2004).

2.2 Physical Limitations

The DBM in the form presented in this paper is a model which considers the ICME as an expanding body that propagates through an isotropic environment that spreads out at a constant speed. The 1/r 2 fall-off of the ambient density, the assumption that the effective ICME cross-section increases as A∝r 2, and the assumption of constant mass of ICME implies a constant value of the drag parameter γ. However, this assumption is valid only if also the dimensionless drag coefficient c d is constant over the considered distance range. Yet, this might not be true (see, e.g., Cargill, 2004), and thus the constant-γ approximation intrinsically affects the accuracy of the arrival-time and impact-speed predictions. Note that constant-γ approximation might also not be appropriate in the case of very tenuous ICMEs (see Section 2.1). Finally, although the approximation A∝r 2 seems to be valid at typical coronagraphic distances, since the CME angular width remains constant, this might not be true at larger heliospheric distances. For example, Bothmer and Schwenn (1998) have found from statistical analysis of the Helios, Voyager, and Pioneer data, measured in the distance range of 0.3 – 4.2 AU, that the radial sizes of ICMEs increase as L∝r 0.78±0.1, whereas the proton density decreases as N∝r −2.4±0.3. Taking approximately N∝1/AL, this indicates that the cross-sectional area increases as A∝r 1.6, thus slower than r 2. However, taking into account the data scatter, the r 2 behavior is still within the error-limits (note also that the approximation N∝1/AL is very crude).

Another intrinsic drawback of the employed model is that the equation of motion contains solely the drag term, i.e., it is assumed that the driving Lorentz force has already ceased at heights below the considered propagation distance range (for a discussion see, e.g., Chen and Kunkel, 2010 and references therein; for the observational aspect, see, e.g., Vršnak, 2001b and Vršnak et al., 2004). However, in some cases this might not be true, since sometimes it is observed that even fast CMEs (i.e., CMEs that are faster than solar wind) still significantly accelerate beyond R=20 (see, e.g., Vršnak et al., 2004). Certainly, such a prolonged action of the Lorentz force can lead to a wrong prediction of the arrival time and impact speed. This can be avoided by employing kinematical measurements based on STEREO observations, i.e., by using a larger take-off distance r 0, where the Lorentz force becomes definitely negligible (Vršnak et al. 2004).

Finally, the model considers a simplified background solar-wind structure, i.e., it is assumed that all parts of the ICME are embedded in an isotropic flow, where the flow speed does not change with distance. However, during the ICME propagation the ambient regime might change, e.g., a fast ICME might first propagate through the slow solar wind and then it can enter a fast solar-wind stream (for an analysis of such events see Temmer et al., 2011). Furthermore, fast ICMEs can encounter slow ICMEs that were launched earlier in the same direction, or slow ICMEs might be “pushed” by fast ICMEs that were launched later (for the application of the model to such an event see Temmer et al., 2012). In this respect also note that ICMEs are spatially quite extended objects, so most likely, there will always be a certain influence of the high-speed wind originating from polar coronal holes (see, e.g., Odstrcil, Riley, and Zhao, 2004).

2.3 Model Input/Output

The basic observational input parameters for DBM are related to the coronagraphic observations of CMEs. In particular, it is required to specify the velocity v 0 when the CME was located at a given distance r 0. Preferably, r 0 should be around, or beyond, a radial distance of r=20r ⊙, so that the conditions γ=const. and w=const. are approximately fulfilled.

Note that measurements of both r 0 and v 0 are burdened by projection effects (e.g., Burkepile et al., 2004; Schwenn et al., 2005; Vršnak et al., 2007; Michalek, Gopalswamy, and Yashiro, 2009). There are various methods which can improve, to a certain degree, the accuracy of the estimates of r 0 and v 0 (e.g., Schwenn et al., 2005; Xie, Ofman, and Lawrence, 2004 and references therein). The validation analysis presented in Section 3 shows that the procedure proposed by Schwenn et al. (2005) gives the best results in the statistical sense. However, the results are only slightly better than those obtained by applying plane-of-sky values for r 0 and v 0. Bearing in mind other uncertainties (Sections 2.2 and 3) as well as the accuracy of measurements, this implies it is sufficient to use the plane-of-sky values for the DBM input.

To complete the set of necessary input parameters, the drag parameter γ and the solar-wind speed w have to be specified. From the physical point of view, the value of γ can be estimated, e.g., by using Equation (3). As an example, let us consider a CME that is several times denser than the surrounding corona, say ρ/ρ w=5, and that its radial thickness is somewhere between 1 and 10 solar radii, corresponding to thin up to thick flux-rope regimes. Substituting these values into Equation (3), one finds γ∼2 – 0.2×10−7 km−1, respectively. Most reasonable combinations of ρ/ρ w and L would fall within or close to this range. This range is consistent with the outcome of a statistical analysis of transit times applied to a set of CMEs presented in Section 3. The analysis shows that from a statistical point of view γ most often attains values in the range 2×10−8 – 2×10−7 km−1.

To conclude, in the case of massive ICMEs (generally meaning bright CMEs in the coronagraphic images) γ should have a small value on the order of 10−8 km−1, whereas in the case of low-density ICMEs (dim in coronagraphic images) it should be closer to the upper limit, i.e., 2×10−7 km−1.

Considering the solar-wind speed, it is important to note that, generally, it depends on the location and time. However, in the simplest form of the DBM, we assume that the solar-wind speed is isotropic and constant. Thus, the straightforward option would be to use a typical slow solar-wind speed of 400 km s−1, or somewhat lower, say 300 km s−1 in the period of deep solar minimum. Yet, if there is an equatorial coronal hole in the vicinity of the ICME source region, one should apply a higher value, say 500 – 600 km s−1, since most likely, the ICME would be propagating at least partly through a high-speed solar-wind stream (HSS). In such a case, a high value of the solar-wind speed should be combined with a low value of γ since HSSs are characterized by low density. To check the possibility for interaction of the ICME with a HSS, it would be good to consult some of the numerical models that simulate the background solar wind (for details, see Temmer et al., 2011).

According to the statistical analysis presented in Section 3, the most appropriate input values for the solar-wind speed should be within the range 300 – 600 km s−1, and in the statistical sense, the best results are obtained for w=500 km s−1. In this respect, let us note that we have performed also a similar statistical analysis where we used the solar-wind speed based on the in-situ measurements at the time of the ICME take-off. This approach resulted in practically the same (even somewhat worse) statistical outcome regarding the accuracy of “predicted” transit times.

To illustrate the effect of different input values for γ and w, let us consider as an example an ICME which had a take-off speed of v 0=1000 km s−1 at R 0=20. The DBM-results for different combinations of γ and w are displayed in Table 1 (see also Figure 1). The first three rows show the results for w=500 km s−1, and different values of γ. Inspecting the results, one finds an uncertainty for the travel time of δT∼± 5 h, and for the impact speed δv 1∼± 30 km s−1. In the bottom three rows, the outcome for γ=1×10−7 km−1 and various values of w is shown. For the travel time, one finds an uncertainty of δT∼± 10 h, and for the impact speed δv 1∼± 100 km s−1. The presented examples illustrate how important it is to check if there was a coronal hole in the vicinity of the CME source region, i.e., if the ICME motion would be affected by a high-speed solar-wind stream.

Table 1 Examples illustrating how the choice of parameter γ and the solar-wind speed w affects the DBM-calculated transit time T and the impact speed v 1.
Full size table

Note that the output values, i.e., the Sun–Earth transit time T and the 1 AU speed v 1, concern the front boundary of the ejection (for the relation between white-light and in-situ observations and related nomenclature see Rouillard, 2011), i.e., the ICME-associated shock should arrive several hours before the ejection itself, depending on the size and Mach number of the ejection (see, e.g., Russell and Mulligan, 2002). Furthermore, we emphasize that the present form of DBM does not take into account the direction of the ICME motion, i.e., in the case of flank-encounter the arrival time at the Earth might be delayed for several hours, in extreme cases up to one, or even two days. The corresponding impact speeds for hits by the ejecta flank may be 100 to 200 km s−1 slower than for the apex-hit, if a self-similar expanding circular geometry for the ejection boundary is assumed. Derivations and plots for these corrections and an application of such a geometry to STEREO/SECCHI observations are presented by Möstl and Davies (2012) and Davies et al. (2012). The application of this method to DBM will be presented in a separate paper.

3 Validation

In this section we test the performance of the model. First, in Section 3.1 we apply the model to a statistical sample of CME/ICME pairs, in order to check if the previously estimated ranges of values for w and γ are consistent with observations. Furthermore, we try to infer what combination of w and γ would provide the best prediction characteristics in the statistical sense, since for a given CME it is difficult to obtain a reliable observational input for a direct physics-based estimate of both w and γ. Finally, this section provides an insight into the accuracy of predictions from the statistical point of view. Then, in Section 3.2 we apply the model to several ICMEs to illustrate that the DBM is capable of reproducing the Sun–Earth kinematics of ICMEs, i.e., that calculated “trajectories” are consistent with the observed ones.

3.1 Statistical Approach

In order to test the model from the statistical point of view, we employed the sample of 91 Sun–Earth events prepared by Schwenn et al. (2005) and the 30-event sample prepared by Manoharan (2006). Note that we consider only the arrival of the front boundary of the ejecta (not the ICME-driven shock).

First, we consider analytical solutions of Equation (4), valid in the approximation γ(r)=const. and w(r)=const. and providing explicit expressions for v(t) and r(t), as given by Equations (5) and (6). After substituting v(t)=v(T)≡v 1 and r(t)=r(T)≡r 1, and t=T, we get two algebraic equations with unknowns γ and w. Unfortunately, they cannot be solved analytically, to provide explicit forms for γ and w as a function of the observable parameters r 0, v 0, v 1, and T, i.e., they have to be solved numerically.

Equations (5) and (6) can be rewritten as

$$ \gamma=\frac{v_{0}-v_{1}}{(v_{0}-w)(v_{1}-w)T} $$
(7)

and

$$ \frac{E(w)}{\gamma} + wT + r_{0} - r_{1} =0 , $$
(8)

where we abbreviated

$$ E(w)= \ln\biggl[\frac{ (v_{0}-v_{1} ) (v_{0}-w )}{ (v_{1}-w ) (v_{0}+w )}+1 \biggr] . $$
(9)

After substituting γ from Equation (7) into Equation (8) one finds

$$ F(w) \equiv\frac{(v_{1}-w)(v_{0}-w)T}{v_{0}-v_{1}} E(w) + wT + r_{0} - r_{1} = 0 . $$
(10)

Thus, the specific value w(r 0,r 1,v 0,v 1,T) can be found by identifying w for which F(w)=0. Once the value of w is evaluated, the drag parameter γ can be calculated employing Equation (7).

In Figure 2 we present the distribution of values of the solar-wind speed w and the drag parameter γ, obtained by employing the data provided by Schwenn et al. (2005) and Manoharan (2006). Note that in a certain fraction of cases the solution w(r 0,r 1,v 0,v 1,T) does not exist due to the incompatibility of the input values and the function ln appearing in Equation (9), i.e., for those combinations of v 0 and v 1 for which the term in the square-brackets in Equation (9) is negative for any value of w.

Figure 2
figure 2

Distribution of values of the solar-wind speed w and the drag parameter γ obtained by substituting measurements into Equations (10) and (7), respectively. In the upper panel the distribution of the solar-wind speeds measured by ACE are shown for comparison (red columns in the background). The lowest value of the solar-wind speed in the DBM-distribution is 120 km s−1 and the highest one is 920 km s−1.

Full size image

The mean values for the distributions shown in Figure 2 are γ=(1±0.6)×10−7 km−1 and w=470±190 km s−1. Median values are 0.8×10−7 km−1 and 410 km s−1, respectively. In the upper panel of Figure 2, the inferred distribution of solar-wind speeds is compared with the distribution of “CMEless” solar-wind speeds based on the in-situ measurements in the period of low ICME activity in 2005 (day of the year 25 – 125; for details see Vršnak, Temmer, and Veronig (2007) and Temmer, Vršnak, and Veronig (2007)). The overall pattern of the two distributions is similar, having the majority of speeds in the range 300 – 700 km s−1. However, the DBM-distribution shows an excess in the high-velocity tail of the distribution, and even more pronounced overabundance on the low-velocity side. This “smearing” of the DBM-distribution could be attributed to inaccurate CME input parameters used in inferring the solar-wind speeds. Bearing in mind this drawback of the applied procedure, we conclude that the inferred distribution is quite consistent with the real solar-wind distribution.

Interestingly, no correlation is found between γ and w. This might be a somewhat surprising result since the solar-wind speed and density are anticorrelated, so one would expect also that w and γ∝ρ w (see Equation (2)) are anticorrelated. Most likely, the ρ w(w) anticorrelation is masked by a large range of CME masses involved in the sample as well as a spectrum of CME sizes, implying also wide range of CME densities (see Equations (2) and (3)).

It should be also noted that the distribution of values of γ in Figure 2 is asymmetric, inclined towards the lower-value side. We have checked the CME masses determined from the coronagraphic data (compiled in the online LASCO CME Catalog; http://cdaw.gsfc.nasa.gov/CME_list/ , for details see Yashiro et al. (2004)), and we found that the mean mass of the CMEs employed in the analysis was around four times higher than average (considering the lognormal distribution, the geometrical mean is even seven times larger). This explains the asymmetry since for more massive CMEs/ICMEs, the value of γ becomes lower (Equation (2)). Note that, generally, the distribution of values of γ is consistent with the range of values estimated on the “physics-based” approach in Section 2.

In the next step we used the CME/ICME sample from Schwenn et al. (2005) and calculated the “predicted” arrival time for each event using different combinations of γ and w, in the range from 2×10−8 to 2×10−7 km−1 and from 300 to 700 km s−1, respectively. Then, the calculated travel times (C) were compared with the observed ones (O), and the difference O−C was determined. Note that we consider only the arrival of the ICME leading edge (i.e., not the shock).

For each of the applied combinations of γ and w we checked the distributions of O−C values, to find those combinations for which the average value of O−C becomes zero. The results are summarized in Figure 3 where we show median values, average values and standard deviations (from top to bottom, respectively) of the O−C distributions as a function of γ for different values of w. In the left panels mean radial distance and speed (R m,V m) of the CME in the LASCO field-of-view are used as the input values. The right panels show the outcome for the radial distance and speed at the moment of the last LASCO observation (R e,V e).

Figure 3
figure 3

Analysis of the difference between observed and calculated travel times (O−C). In the left panels mean radial distance and speed (R m,V m) of the CME in the LASCO field-of-view are used as input values. The right panels show the outcome for the radial distance and speed at the moment of the last observation (R e,V e). The top panels represent median values, middle panels average values, and the bottom panels standard deviations. The applied values of the solar-wind speed w, expressed in km s−1, are written next to the curves.

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Inspecting Figure 3 one finds that average or median values of O−C become zero for different combinations of γ and w. For example, considering the “R m,V m” option, one finds that if w=500 km s−1 is assumed, the average O−C becomes zero for γ∼0.5×10−7 km−1 (see middle-left panel). Assuming w=400 km s−1 one gets γ∼0.15×10−7 km−1, whereas assuming w=600 km s−1 the corresponding value of γ becomes much higher than values allowed by the “physics-based” range of values estimated in Section 2. On the other hand, note that for, e.g., w=500 km s−1, a combination with any γ in the range ∼ 0.2 – 1.5×10−7 km−1 would result in average O−C in the range ∼± 5 h. Bearing in mind that we used a relatively limited sample of events, and possible statistical fluctuations, all such combinations should also be possible/satisfactory. Thus, the choice of γ does not affect too much the results from a statistical point of view, i.e., the solar-wind speed seems to be the more important parameter. This is consistent with the results presented by Vršnak and Žic (2007), which showed that the ambient solar-wind speed appears to be the most important factor in determining the ICME transit time. The probable reason lies in the fact that in most ICMEs the speeds become close to the solar-wind speed already relatively close to the Sun, whereas very massive ICMEs, where the adjustment to the solar-wind speed is prolonged to larger distances, are relatively rare and do not affect significantly the statistical results.

On the other hand, standard deviations, i.e., the dispersion of the O−C values significantly increases for small values of γ. Standard deviations “stabilize” around the value of 16 h for γ≥10−7 km−1. For γ=10−7 km−1 one finds that the appropriate value of the solar-wind speed is w∼500 – 550 km s−1. If higher values of γ are assumed, also the value of w increases. Note that the combination γ=1×10−7 km−1 and w=500 km s−1 is roughly consistent with the results presented in Figure 2.

The distribution of O−C values for the combination γ=1×10−7 km−1 and w=500 km s−1, for the R e,V e input option, is presented in Figure 4. The extreme delay at the right-hand-side of the distribution shown in the top panel of Figure 4 is almost surely due to a wrong CME/ICME-pair identification, since the delay is larger than two days. The distribution peaks at O−C=0 (as requested), and has a standard deviation of 16 h. The standard deviation decreases to 15 h if the mentioned outlier is excluded.

Figure 4
figure 4

Top: Distribution of O−C values for γ=10−7 km−1 and w=500 km s−1, applied to R e,V e data. Bottom: a corresponding cumulative distribution of |O−C|.

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In the bottom panel of Figure 4 we show the cumulative distribution of |O−C|, where we have excluded the previously mentioned outlier. The mean value of |O−C| equals ∼ 12 h. The distribution shows that ∼ 55 % of events have |O−C|<12 h and more than 85 % of events have |O−C|<1 day. If we restrict the sample only to events where R e>15 or R e>20 the average |O−C| decreases to below 12 h. However, the shape of the cumulative distribution does not change significantly, except that now ∼ 60 % of events have |O−C|<12 h and ∼ 90 % have |O−C|<1 day. In this respect it is important to discuss/resolve what means a “successful prediction”, since the obtained standard deviations of 15 – 16 h, as well as the mean value |O−C| ∼ 12 h, are relatively large. Yet, O−C deviations become significantly smaller when fast CMEs (short transit times) are considered, especially if CMEs are launched from regions close to the solar disc center. Note that this subset of events tends to be more geoeffective, thus for purposes of space-weather forecasting one can count on better accuracy of predictions (this statistically very demanding study will be presented in a separate paper).

The procedure described above was repeated for the sample from Schwenn et al. (2005) using also the mean speed in the LASCO field-of-view (V m) as well as the deprojected value (denoted as “radial” in Schwenn et al., 2005). We also employed the samples from Manoharan (2006) and Zhang et al. (2003). All of these options led to similar results, in statistical sense not significantly different from those presented in Figure 4.

3.2 Tracking the Interplanetary Kinematics

To test the capabilities of DBM in more detail, we have performed a number of case studies, where we compared the deprojected kinematics of ICMEs reconstructed from remote-sensing observations by the STEREO spacecraft with the kinematics calculated by DBM. In Figure 5, the deprojected kinematics of three ICMEs are shown as typical examples. For the first two events (CMEs launched on 15 November 2007 and 12 December 2008, respectively) we employ the measurements presented by Liu et al. (2010). The November 2007 event was analyzed also by Rouillard et al. (2010) and Farrugia et al. (2011), and the December 2008 event by Davis et al. (2009). For the third event, tracked by the STEREO spacecraft over the full Sun–Earth distance range (CME launched on 1 June 2008) we use the measurements presented in detail by Rollett et al. (2012) and Temmer et al. (2011).

Figure 5
figure 5

Heliospheric kinematics of three ICMEs, launched on 15 November 2007 (top), 12 December 2008 (middle), and 1 June 2008 (bottom). Left panels show the deprojected distance versus time, whereas right panels show the deprojected speed versus distance. Observations are drawn in gray, whereas the red line shows the calculated kinematics (parameters are written in the legend). Blue squares in the bottom graphs represent in-situ measurements.

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The ICME of 15 November 2007 was characterized by a low take-off speed, and was continuously accelerating towards the velocity of 500 – 600 km s−1. The DBM kinematics reproduces the observations in the best way by applying the asymptotic solar-wind speed of w=600 km s−1 and γ=1.6×10−7 km−1. Indeed, the CME was launched from a region surrounded by an equatorial coronal hole that was a source of a high-speed solar-wind stream. Furthermore, in the SoHO/LASCO catalogue ( http://cdaw.gsfc.nasa.gov/CME_list/ , see Yashiro et al., 2004) the event was classified as “Poor Event”, implying that this was a low-mass CME, explaining the rather large value of γ.

The ICME of 12 December 2008 was accelerating rapidly up to R∼23, to achieve a relatively high take-off speed of v∼740 km s−1, after which it started to decelerate. The decelerated propagation can be reproduced very closely by DBM, applying γ=2.0×10−7 km−1 and w=350 km s−1. Again, the event was classified in the SoHO/LASCO catalogue as “Poor Event”, consistent with a relatively high value of γ. There were no equatorial coronal holes in the vicinity of the eruption, which explains the low asymptotic solar-wind speed w, and is in addition consistent with the high value of γ (slow solar wind is characterized by denser plasma flow). Note that in this event it would not be possible to predict the ICME arrival based on the LASCO data, since the CME was gradually accelerating up to large heights, whereas in LASCO the CME was followed only up to R∼12, where it had the speed of only ∼ 320 km s−1. Thus, whenever possible, it would be the best to use the deprojected speeds from STEREO as input.

The propagation of the third ICME, launched on 1 – 2 June 2008, was followed remotely beyond 1 AU, and was recorded in situ by STEREO-B (for details see Möstl et al., 2009; Rollett et al., 2012, and Temmer et al., 2011). The event was first studied by Robbrecht, Patsourakos, and Vourlidas (2009) who classified it as “stealth CME” having no recognizable signatures of associated low-coronal activity. Various aspects of the event were also analyzed by Möstl et al. (2009), Lynch et al. (2010), and Wood, Howard, and Socker (2010). The remotely measured heliospheric kinematics presented in Figure 5 is based on the harmonic mean method (Howard and Tappin 2009; Lugaz, Vourlidas, and Roussev 2009), modified in such a way as to follow the ICME segment that propagates towards the in-situ observer (for details see Möstl et al., 2010; Rollett et al., 2012; Temmer et al., 2011, and references therein). As seen from the bottom graphs of Figure 5, the DBM with w=440 km s−1 and γ=2×10−7 km−1 reproduces quite well the remotely measured kinematics as well as the in-situ measurements. The applied solar-wind speed is consistent with the modeled solar-wind structure (for details see Temmer et al., 2011). Furthermore, the LASCO data show that the CME was rather dim, consistent with the relatively high value of γ.

Finally, we emphasize that the relationship between the white-light observations and the in-situ measurements is not trivial (Rouillard 2011). In fast/bright ICMEs, the white-light leading edge probably depicts the compressed shock-sheath region, whereas in slow shockless ICMEs the situation is less clear. Here, we have adjusted the DBM input parameters to fit the kinematics of the ICME leading edge, assuming that the ICME expands in a self-similar manner and that the sheath region is not very thick, i.e., that the propagation of the white-light leading edge is not much different from the propagation of the ejection itself. Of course, this is a very crude approximation, but our intention was just to illustrate that the overall shape of kinematical curves can be reproduced well by the DBM. A more detailed analysis of the herein employed ICMEs and several other events will be presented in a separate paper.

4 Conclusion

In this paper we presented a version of the drag-based model that provides analytical solutions for the ICME kinematics. This allows easy handling and prompt application in the real-time space-weather forecasting (see the Appendix). The model can be adjusted to various situations, i.e., provides distinction between weak-drag or strong-drag effects as well as choosing between slow and fast solar-wind environment. However, in the present state it cannot account for ICME–ICME interactions, though it can help in the analysis and comprehension of such events (for an example see Temmer et al., 2012). Another drawback is that using this (the simplest) version of the model, only the arrival of the front boundary of the ejecta can be predicted, i.e., it does not account for the impact of the ICME-driven shock.

The application of the model to a statistical sample of events revealed that the drag parameter γ most often ranges between 2×10−8 km−1 and 2×10−7 km−1, consistent with the physics-based estimates. The lower values are appropriate for massive ICMEs in fast solar-wind environment, characterized by a low density. High values apply to low-density ICMEs in the slow solar wind. This effect explains the asymmetry present in Figure 2, since the mean mass of the CMEs from the employed sample was considerably higher than average. The optimum value of the solar-wind speed, w∼500 km s−1, can be explained by a large spatial extent of ICMEs, so their kinematics are at least partly affected by the fast wind from polar coronal holes. Furthermore, at least a fraction of ICMEs propagate interacting with high-speed solar-wind streams from equatorial coronal holes, which also increases the average value of w.

The validation analysis showed that the expected typical errors in predicting the arrival times of ICMEs by the presented version of the model is around 0.5 days. We believe that the distribution of errors can be significantly reduced by taking the previously mentioned drawbacks into account. We will focus on these aspects when developing a more advanced version of the model. Finally, we demonstrated that the measured heliospheric ICME kinematics can be reproduced by the model, at least for isolated events propagating in a simple solar-wind environment. Yet, as shown by Temmer et al. (2011, 2012) the ICME propagation can be much more complex in situations when the ambient solar-wind regime changes, or for the events which include ICME–ICME interaction.