© D. Sudar et al. 2022

1. Introduction

The drag-based model (DBM, Vršnak et al. 2013; Žic et al. 2015) is a very popular tool for analysing the propagation of coronal mass ejections (CMEs) through interplanetary space. Its roots can be traced to observational facts about kinematic observations in the vicinity of the Sun (Vršnak et al. 2004), which found that CMEs with speeds higher than the ambient solar wind decelerate on average, while those slower than the ambient solar wind speed accelerate. Similar observations also have been made by Gopalswamy et al. (2000, 2001), Yashiro et al. (2004), Manoharan (2006), Tappin (2006), Vršnak & Žic (2007), and Vršnak et al. (2008). Following the work and magnetohydrodynamics (MHD) simulations performed by Cargill (2004), Vršnak et al. (2013) presented the DBM where the drag force has a quadratic dependence on the CME speed relative to that of the ambient solar wind. This leads to a set of kinematic equations that resemble aerodynamic drag. This does not imply that DBM represents the aerodynamic drag in interplanetary space, but rather is an analogue of the aerodynamic drag in the Earth’s atmosphere. An analysis of the MHD simulations of a flux rope travelling through interplanetary space and the relation to drag is given in Cargill (2004) and references therein. Another observational support of the drag-like behaviour of CMEs in interplanetary space was presented by Sudar et al. (2016). The authors found that the plot of observed CME transit times to Earth versus initial CME speed (their Fig. 8) resembles the DBM model curves shown in Fig. 5 presented in Vršnak & Žic (2007).

The initial DBM model only considered the one-dimensional case that ignores the shape of the CME leading edge, which in reality affects the transit times depending on whether the CME hits the target (e.g., Earth) with its apex or closer to its flank. Žic et al. (2015) expanded the DBM model to two dimensions. The idea was that the initial shape of the CME leading edge is a semicircle spanning over the full width of the CME, where the width remains constant throughout the entire trajectory. This means that each element of the CME leading edge starts at a different height above the solar surface, which is effectively a function of the angular distance of the element from the apex of the CME leading edge. The same angular function was used to determine the initial speed of each element. After the eruption, each leading-edge element moves independently according to the DBM equations. Žic et al. (2015) also discussed the leading edge of a CME in later stages of the propagation and suggested that the leading edge flattens. Isavnin (2016) also mentioned a flattening of the CME leading edge for CMEs travelling faster than the solar wind. They referred to the DBM model. Similar statement can be found in Dumbović et al. (2018a,b) or Kilpua et al. (2019) in reference to the Žic et al. (2015) paper. More recently, Dumbović et al. (2021) stated that the radial difference between the CME apex and its flank increases with time and converges to a constant value, but they did not offer mathematical proof.

This paper is intended to clarify what happens with the shape of the CME leading edge and and how it behaves at the asymptotic limit within the framework of the DBM equations. We also investigate the velocity profile of the leading edge as it evolves with time.

There are other models of CME propagation that also use the quadratic drag equation to describe, at least in part, the kinematic of the CME (e.g., Möstl et al. 2015; Rollett et al. 2016; Kay & Gopalswamy 2018). The differences of these models to the DBM range from different initial CME geometry and speed profiles to the inclusion of other forces that can impact the arrival time of the CME front at the target. The impact of the analysis presented in this paper on these other models is briefly discussed as well.

2. Definitions of the DBM

We start with the analytical solution of the one-dimensional model, which can be obtained by some simplifications of the problem (Vršnak et al. 2013). The DBM solves the quadratic drag equation,

$$\begin{array}{c}\hfill a(t)=-\gamma (v-w)|v-w|,\end{array}$$(1)

where a is the force per unit mass, γ is the drag parameter, v is the current speed of the CME, and w is the speed of the solar wind. In this equation, the drag parameter, γ, and the solar wind speed, w, are considered constant (Vršnak et al. 2013). For clarity, we repeat the solution for the speed, v, and the radial distance, r, with respect to time, t,

$$\begin{array}{cc}& v(t)=\frac{v_0-w}{1\pm \gamma (v_0-w)t}+w,\hfill \end{array}$$(2)

$$\begin{array}{cc}& r(t)=\pm \frac{1}{\gamma }ln[1\pm \gamma (v_0-w)t]+wt+r_0,\hfill \end{array}$$(3)

where the plus sign is chosen when v₀ >  w and the minus sign when v₀ <  w. r₀ is the initial distance, w is the speed of the solar wind (assumed constant), v₀ is the CME initial speed, and γ is the so-called drag parameter that describes the strength of the interaction between the CME and the ambient solar wind. This equation can be used to calculate the CME transit time to a desired target at some distance, r, from the Sun.

Expanding the description to two dimensions, we define the initial shape of the CME leading edge at the start according to Žic et al. (2015) with

$$\begin{array}{c}\hfill r(\varphi )=r_0\frac{cos\varphi +\sqrt{{tan}^2\omega -{sin}^2\varphi }}{1+tan\omega }=r_{0}F(\varphi ),\end{array}$$(4)

where ϕ is the angular distance from the apex of the CME, and ω is its half-width. The equation describes a semicircle embedded on top of the CME spanning its full width (2ω). Žic et al. (2015) also provided the equation for the speed profile of the CME leading edge, which is characterised by the same angular function F(ϕ),

$$\begin{array}{c}\hfill v(\varphi )=v_{0}F(\varphi ),\end{array}$$(5)

which can be obtained by differentiating Eq. (4) with respect to time. The given speed equation (Eq. (5)) implies self-similar expansion of the CME, that is, the shape of the CME leading edge is preserved in time (Vršnak et al. 2019). This leads us to the first ambiguity of the way in which the CME evolves after the drag becomes dominant, in particular its shape. If we impose that the shape (Eq. (4)) is preserved, all we need to do is to calculate the radial distance of the centre (Eq. (3)) and multiply it by the angular function, F(ϕ), to obtain the radial distance of any element of the leading edge at the angular position ϕ from the centre.

Another possibility is that the initial radial distance and speed are given by Eqs. (4) and (5), but elements of the CME move independently of each other, where each elements moves according to Eq. (3) further on. We can even provide an analytical solution for each element with the following equation:

$$\begin{array}{c}\hfill r(\varphi ,t)=\pm \frac{1}{\gamma }ln[1\pm \gamma (v_{0}F(\varphi )-w)t]+wt+r_{0}F(\varphi ),\end{array}$$(6)

where each element is uniquely identified by ϕ. The positive sign, the plus, should be used when v₀F(ϕ)> w, and the minus sign when v₀F(ϕ)< w.

This latter possibility is currently used in the DBM tool, and we therefore discuss in the following sections the evolution of the latter model (Eq. (6)). The evolution of the CME leading edge in this model is sometimes described as flattening. While it is true that the curvature of the leading edge is smaller than in the first model (Eq. (4)), the word flattening causes some confusion. Sometimes it is mistakenly thought that the leading edge assumes the form of a circle centred around the Sun. We show below that this is not the case, and that some curvature still remains even as time approaches infinity.

3. Asymptotic behaviour of the CME leading edge

For the CME leading edge to become part of a circle centred around the Sun, its shape must satisfy the equation

$$\begin{array}{c}\hfill \frac{\partial r(\varphi ,t)}{\partial \varphi }{|}_{t\to \infty }=0,\end{array}$$(7)

as time, t, approaches infinity.

First we define F′(ϕ)=dF(ϕ)/dϕ as a derivative of the shape function F(ϕ) with respect to ϕ, which gives the following expression:

$$\begin{array}{c}\hfill F^{\prime }(\varphi )=\frac{\mathrm{d}F(\varphi )}{\mathrm{d}\varphi }=-\frac{\frac{cos\varphi sin\varphi }{\sqrt{{tan}^2\omega -{sin}^2\varphi }}+sin\varphi }{1+tan\omega }.\end{array}$$(8)

Differentiating Eq. (6) with respect to ϕ, we obtain

$$\begin{array}{c}\hfill \frac{\partial r(\varphi ,t)}{\partial \varphi }=\frac{t{v}_0{F}^{\prime }(\varphi )}{\pm t\gamma (v_{0}F(\varphi )-w)+1}+r_0{F}^{\prime }(\varphi ).\end{array}$$(9)

As time becomes very long, t → ∞, we can neglect the term +1 in the denominator, which does not grow with t, and we obtain

$$\begin{array}{c}\hfill \frac{\partial r(\varphi ,t)}{\partial \varphi }{|}_{t\to \infty }\approx \pm \frac{v_0{F}^{\prime }(\varphi )}{\gamma (v_{0}F(\varphi )-w)}+r_0{F}^{\prime }(\varphi ).\end{array}$$(10)

In Fig. 1 we show the $\frac{\partial r(\varphi ,t)}{\partial \varphi }$ function for several time instants. We used generic values γ = 1 km⁻¹, r₀ = 1 km, w = 1 km s⁻¹, v₀ = 2 km s⁻¹, and ω = 60^° , which represents a wide and fast CME. In Fig. 2 we show the same function, but this time, for a CME that is slower than the solar wind speed (v₀ = 0.75 km s⁻¹). The functions are shown for time instances t = 1 s, t = 10 s, and t = 100 s with solid red, green, and blue lines, respectively. In addition, we show the asymptotic curve, Eq. (10), as a solid thick grey line. The shape converges to the asymptotic value, which is not equal to zero, except for the apex (ϕ = 0). For negative values of ϕ, $\frac{\partial r(\varphi ,t)}{\partial \varphi }$ is positive, which means that as we move from the left flank (ϕ <  0) towards the apex (Δϕ >  0), the radial distance grows (Δr >  0). For positive values, the opposite is true. Farther away from the apex towards the right flank (Δϕ >  0), the radial distance drops (Δr <  0). This shows in both cases that radial distance of points closer to the apex is always larger than for those farther away from the apex.

Fig. 1. Time evolution of the $\frac{\partial r(\varphi ,t)}{\partial \varphi }$ function. The thick grey line shows the asymptotic curve (Eq. (10)), and the red, green, and blue lines show the curve from Eq. (9) at various time instants shown on the plot. All parameters are equal to one, except for the half-width, ω = 60°, and initial speed, v0 = 2 km s−1, representing a CME travelling faster than the solar wind.

Fig. 2. Same as Fig. 1, but the initial speed is v0 = 0.75 km s−1, representing a CME slower than the solar wind.

The variations in $\frac{\partial r(\varphi ,t)}{\partial \varphi }$ in Figs. 1 and 2 are of the order of unity because all factors in Eq. (10) are close to unity due to our choice of generic values. In the Solar System, the value of γ is about 10⁻⁷ km⁻¹ so that 1/γ in the first term of Eq. (10) is of the order of 10 R_⊙. The initial distance of the apex, r₀, is also typically of the same order of magnitude. This means that the change in radial distance per radian is of the order of 10 R_⊙. To derive the change in radial distance along the leading edge of the CME per degree, we need to multiply by the factor π/180 ≈ 1/60, which gives the change of the radial distance to a fraction of the solar radius per degree. The time instances indicated in Figs. 1 and 2 would be in the order of hours to hundreds of hours.

We have shown that the apex, ϕ = 0, of the CME always remains in front of the rest of the structure. We can therefore investigate how far the flank of the CME, ϕ = ω, trails behind. In order to do this, we define the difference

$$\begin{array}{c}\hfill \mathrm{\Delta }r(t)=r(0,t)-r(\omega ,t),\end{array}$$(11)

which we evaluate by using the Eq. (6) for ϕ = 0 and ϕ = ω. We obtain

$$\begin{array}{c}\hfill \mathrm{\Delta }r(t)=\pm \frac{1}{\gamma }ln\frac{1\pm \gamma (v_0-w)t}{1\pm \gamma (v_{0}F(\omega )-w)t}+r_0(1-F(\omega )),\end{array}$$(12)

where we used the fact that F(0)=1. F(ω) can be simplified to 1/(cosω + sinω). The asymptotic behaviour can be obtained by letting t → ∞, which leads to a constant term

$$\begin{array}{c}\hfill \mathrm{\Delta }r(t){|}_{t\to \infty }\approx \pm \frac{1}{\gamma }ln\frac{v_0-w}{v_{0}F(\omega )-w}+r_0(1-F(\omega )).\end{array}$$(13)

To illustrate the behaviour of Δr(t), we created four examples with the realistic inputs listed in Table 1. Two models, HH and HL, use a high value of the drag parameter γ = 2.0 × 10⁻⁷ km⁻¹, while the other two models have lower γ = 0.5 × 10⁻⁷ km⁻¹. The HH and LH models have an initial CME speed of v₀ = 1000 km s⁻¹ , which is higher than the solar wind speed, w = 500 km s⁻¹. The other two models, HL and LL, simulate CMEs that are slower than the solar wind speed with v₀ = 300 km s⁻¹.

In Fig. 3 we show the evolution of Δr(t) with time for the four models listed in Table 1. We also show the locations of the planets when the apex reaches them for each model in the same figure. For high γ, the asymptotic constant form of Δr(t) is reached well before the Earth’s orbit, while for low γ, the difference, Δr(t), still grows slowly even up to the Jupiter orbit.

Fig. 3. Δr(t) is shown as a function of time, t, for four different models from Table 1. The empty squares, empty triangles, empty circles, empty diamonds, and empty pentagons show the position when the apex of the CME reaches Mercury, Venus, Earth, Mars, and Jupiter, respectively.

The velocity profile for any ϕ and time t can be obtained by combining Eqs. (2) and (5), which yields

$$\begin{array}{c}\hfill v(\varphi ,t)=\frac{v_{0}F(\varphi )-w}{1\pm \gamma (v_{0}F(\varphi )-w)t}+w.\end{array}$$(14)

The asymptotic behaviour when t → ∞ gives that velocity for each segment ϕ converges to the speed of the solar wind, v(ϕ)=w. Because v is no longer a function of ϕ as t → ∞, we immediately obtain that $\frac{\partial v(\varphi ,t)}{\partial \varphi }\to 0$ and Δv(t)=v(0, t)−v(ω, t)→0 as t → ∞.

Two examples of velocity profile convergence with generic values γ = 1 km⁻¹, r₀ = 1 km, w = 1 km s⁻¹, and ω = 60^° are shown in Fig. 4. One example illustrates a CME that is faster than the solar wind with v₀ = 2 km s⁻¹, and the other example shows a CME with an initial speed lower than the solar wind speed, v₀ = 0.7 km s⁻¹. The asymptotic value of v(ϕ, t)=w = 1 km s⁻¹ is shown with the solid thick grey line. The time evolution of a CME with v₀ = 2 km s⁻¹ is shown with three solid lines above the grey line, while for the slower CME, the three curves are shown with dashed lines below the asymptotic grey line.

Fig. 4. Asymptotic behaviour of the velocity profile, v(ϕ, t). The thick solid grey line represents the solar wind speed, w, and the asymptotic profile of v(ϕ, t). Two models for three instants in time are shown with red, green, and blue curves. The model with a higher initial speed than the solar wind speed is shown with three solid coloured curves above the grey asymptotic line, and the curves of the model with v0 = 0.75 km s−1 <  w are shown with dashed lines below the grey line.

To obtain plausible values for the solar system in Fig. 4, it is sufficient to take note of the asymptotic value of v(ϕ, t)=w in Eq. (14), which is of the order of 100 km s⁻¹ in the heliosphere, and to change the scale of the vertical axis by the same number. Time instances corresponding to the curves shown in Fig. 4 would be from the order of hours to hundreds of hours.

In Fig. 5 we show the a(v) function from the Eq. (1) for three different drag parameters: γ₁ = 2 × 10⁻⁷ km⁻¹, γ₂ = 1 × 10⁻⁷ km⁻¹, and γ₃ = 0.5 × 10⁻⁷ km⁻¹ with solid lines. The value of the solar wind speed is w = 500 km s⁻¹ for all three cases. For each γ, we also show the location of three characteristic CMEs. For the CME denoted Slow, both the initial apex and the flank speed are slower than w = 500 km s⁻¹. For the Fast CME, the speeds of the apex and the flank are faster than w, and in the Mixed case, the initial speed of the apex is higher than w, but the speed of the flank is lower than w. Apexes and flanks are shown with filled and empty symbols, respectively.

Fig. 5. a(v) relation shown with solid lines for γ1 = 2 × 10−7 km−1, γ2 = 1 × 10−7 km−1, and γ3 = 0.5 × 10−7 km−1. The initial locations of apexes and flanks for three classes of CMEs are plotted as well (squares: slow CME; circles: mixed CME; triangles: fast CME). Flanks are always shown with empty symbols, and their corresponding apexes are plotted with filled symbols of the same type.

The apexes are always farther to the right than their flanks because their initial speed is always higher. All the points will move towards the equilibrium point, a = 0 and v = w = 500 km s⁻¹, as time progresses. Each point will always remain on its γ curve as it moves towards equilibrium. For the Slow CMEs, the points of the apexes and flanks will move to the right as the CME is accelerated, but flanks cannot overtake the apexes. Similarly, for Fast CMEs, the points will move to the left due to deceleration, but the apexes will never move further left than their corresponding flanks. Perhaps the most illustrative example is the Mixed case, where apexes move to the left towards the equilibrium, while the flanks move to the right. It is obvious that in this case, the speed of the flank cannot ever become higher than the speed of its corresponding apex. Moreover, in the Mixed case, parts of the CME leading edge move according to Eq. (6) with the plus sign when v₀F(ϕ)> w and the other parts with the minus sign when v₀F(ϕ)< w.

4. Discussion and conclusion

We have shown that the leading edge of the CME never becomes fully circular and that $\frac{\partial r(\varphi ,t)}{\partial \varphi }$ never becomes zero for all ϕ. Moreover, the distance between the apex, r(0), and the flank, r(ω), monotonically increases with time up to the asymptotic value. We can generalise this statement for any ϕ, so that for any two segments for which ϕ₁ >  ϕ₂, it follows that the difference is r(ϕ₂)−r(ϕ₁)> 0 and it monotonically grows with time and becomes constant as t → ∞. Similarly, for speed, we can generalise that v(ϕ₂)−v(ϕ₁)≥0, but the difference becomes smaller with time and reaches zero at infinity.

The term flattening has some meaning only when the self-similar expansion model from Eq. (4) is compared with the model we analysed in this paper, where each segment of the CME propagates independently, as given in Eq. (6). In this comparison, the leading edge of the self-similar expansion model is more curved, so that flattening refers to the less pronounced curvature in the other model. Another source of confusion is that the whole structure, characterised by r, grows with time and the difference Δr(t) seems less pronounced because it grows more slowly with time and approaches the finite limit, while r grows to infinity. However, we recall that Δr(t) does not become smaller with time; the opposite is true.

In the quadratic drag model used in DBM, there is simply no mechanism that will accelerate an initially slower segment at a smaller distance to speeds higher than the speed of the apex, which is a necessary requirement for the segment to catch up with the apex. Except for modifying the initial geometry and speed profile of the CME, the only options left to propel the flank segments to speeds higher than that of the apex is to make the drag parameter, γ, or solar wind speed, w, or both, functions of ϕ. In this case, the apex of the CME and its flank would lie on different curves in Fig. 5. In this scenario, the flank could in some cases overtake the apex in finite time. It is also possible that γ and w are functions of time. This would mean that different elements experience different drag force simply because they arrive at a different distance, r, at different times. This would have similar consequences as the angular, ϕ, dependence of the solar wind parameters discussed above.

For other models that describe the kinematics of the CME with Eq. (1), we can also conclude the same. What was initially slower and farther behind will always remain behind with increasing distances as long as the drag parameter, γ, and/or solar wind speed are solely a function of distance, r, or constant. Recent results that showed that CMEs are not fully rigid structures (see e.g., Owens et al. 2017) allow for the possibility that the anisotropic solar wind affects different segments differently. One of the implicit assumptions in the solution given in Eq. (6) is that each element moves independently. This assumption is partly but not fully justified by Owens et al. (2017). Some interaction between adjacent elements, such as magnetic tensions, does exist.

As discussed above, the angular dependence of γ and w can lead to deformations of the leading edge. This possibility was used by Hinterreiter et al. (2021) and resulted in a deformed front (their Fig. 3) and a speed profile that converged towards the anisotropic solar wind speed (their Fig. 6). The models that include additional forces, such as Kay & Gopalswamy (2018), complicate the situation and it is not straightforward to apply the same reasoning, but it is worthwhile to know that effects such as deformations and catching up with faster elements occur either because of anisotropy or time dependence of the solar wind parameters or because of the effects of other forces.

Acknowledgments

This work has been supported by the Croatian Science Foundation under the project 7549 “Millimeter and submillimeter observations of the solar chromosphere with ALMA”.

References

All Tables

All Figures

Fig. 1. Time evolution of the $\frac{\partial r(\varphi ,t)}{\partial \varphi }$ function. The thick grey line shows the asymptotic curve (Eq. (10)), and the red, green, and blue lines show the curve from Eq. (9) at various time instants shown on the plot. All parameters are equal to one, except for the half-width, ω = 60°, and initial speed, v0 = 2 km s−1, representing a CME travelling faster than the solar wind.

In the text

	Fig. 2. Same as Fig. 1, but the initial speed is v0 = 0.75 km s−1, representing a CME slower than the solar wind.
In the text

	Fig. 3. Δr(t) is shown as a function of time, t, for four different models from Table 1. The empty squares, empty triangles, empty circles, empty diamonds, and empty pentagons show the position when the apex of the CME reaches Mercury, Venus, Earth, Mars, and Jupiter, respectively.
In the text

Fig. 4. Asymptotic behaviour of the velocity profile, v(ϕ, t). The thick solid grey line represents the solar wind speed, w, and the asymptotic profile of v(ϕ, t). Two models for three instants in time are shown with red, green, and blue curves. The model with a higher initial speed than the solar wind speed is shown with three solid coloured curves above the grey asymptotic line, and the curves of the model with v0 = 0.75 km s−1 <  w are shown with dashed lines below the grey line.

In the text

	Fig. 5. a(v) relation shown with solid lines for γ1 = 2 × 10−7 km−1, γ2 = 1 × 10−7 km−1, and γ3 = 0.5 × 10−7 km−1. The initial locations of apexes and flanks for three classes of CMEs are plotted as well (squares: slow CME; circles: mixed CME; triangles: fast CME). Flanks are always shown with empty symbols, and their corresponding apexes are plotted with filled symbols of the same type.
In the text