AN ANALYTICAL MODEL OF INTERPLANETARY CORONAL MASS EJECTION INTERACTIONS

Before the CME launch, at quiet times, the Sun blows out a stationary wind with velocity v₀, density ${\rho }_{0}$, and mass-loss rate $\dot{{m}_{0}}$ for injection times $\tau \lt 0.$ Then, at $\tau =0,$ the first perturbation (CME₁) starts flowing with velocity ${v}_{1}={a}_{1}\;{v}_{0},$ density ${\rho }_{1}={c}_{1}\;{\rho }_{0}$, and mass-loss rate $\dot{{m}_{1}}={b}_{1}\;\dot{{m}_{0}}$ lasting for a time interval ${\rm{\Delta }}{\tau }_{1}$ (Figure 2). Following the formalism of Cantó et al. (2000; see also González & Cantó 2002; Cantó et al. 2005), it can be shown that the first WS₁ is instantaneously formed when the jump of the flow parameters occurs. Then, the dynamical evolution of the shocked material depends on the jump factors a₁, b₁, and ${c}_{1}(={b}_{1}/\;{a}_{1}).$ The evolution of the WS₁ comprises two phases, A₁ and B₁.

2.1.1. Phase A₁

The initial phase A₁ is characterized by a constant velocity with an intermediate value between the ASW and the CME₁ velocities. This is due to the fact that both flows are transferring momentum to the WS₁ at constant rates. Therefore, the initial velocity of the WS₁ depends on the interplay between the momentum injected into the WS₁ by both the CME₁ and the ambient wind ahead.

To solve this phase, we find the function F (see Equation (3) and Figure 2);

$$\begin{eqnarray}F & = & {\displaystyle \int }_{{\tau }_{0}}^{0}[{x}_{s}-{v}_{0}\;(t-\tau )]\;\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{0}^{{\tau }_{1}}[{x}_{s}-{v}_{1}\;(t-\tau )]\;\dot{{m}_{1}}\;d\tau \\ & = & \displaystyle \frac{1}{2}\;\left\{\dot{{m}_{1}}\;{\tau }_{1}\left[2\;{x}_{s}+({\tau }_{1}-2\;t)\;{v}_{1}\right]\right.\\ & & -\left.\dot{{m}_{0}}\;{\tau }_{0}\left[2\;{x}_{s}+({\tau }_{0}-2\;t)\;{v}_{0}\right]\right\},\end{eqnarray} \tag{ 7 }$$

where the first integral is related to the ASW in front of the WS₁ (with ${\tau }_{i}={\tau }_{0},$ ${v}_{i}={v}_{0}$ and ${\tau }_{0}\leqslant 0$), while the second integral corresponds to the CME₁ behind of the WS₁ (with ${\tau }_{j}={\tau }_{1}$ and ${v}_{j}={v}_{1}$ and $0\leqslant {\tau }_{1}\leqslant {\rm{\Delta }}{\tau }_{1}$),

Using the condition F = 0 (see Equation (4)) and the free-streaming conditions (see Equations (5) and (6)),

$$\begin{eqnarray}&&{x}_{s}={v}_{0}\;(t-{\tau }_{0}),\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&{x}_{s}={v}_{1}\;(t-{\tau }_{1}),\end{eqnarray} \tag{ 9 }$$

for this phase, we find,

$$\begin{eqnarray}&&{\tau }_{0}=-\sqrt{{a}_{1}\;{b}_{1}}\;{\tau }_{1},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&t=\left(\displaystyle \frac{{a}_{1}+\sqrt{{a}_{1}\;{b}_{1}}}{{a}_{1}-1}\right){\tau }_{1},\end{eqnarray} \tag{ 11 }$$

and,

$$\begin{eqnarray}&&{x}_{s}=\left(\displaystyle \frac{1+\sqrt{{a}_{1}\;{b}_{1}}}{{a}_{1}-1}\right){a}_{1}\;{v}_{0}\;{\tau }_{1},\end{eqnarray} \tag{ 12 }$$

where we have used ${\tau }_{1}$ as a free parameter.

From Equations (11) and (12) it follows that the WS₁ moves with a constant velocity given by

$$\begin{eqnarray}&&{v}_{s}=\displaystyle \frac{1+\sqrt{{a}_{1}\;{b}_{1}}}{{a}_{1}+\sqrt{{a}_{1}\;{b}_{1}}}\;{a}_{1}\;{v}_{0}\end{eqnarray} \tag{ 13 }$$

during this initial phase. Thus, we may write from Equations (11) to (13), ${x}_{s}={v}_{s}t,$ with v_s given by Equation (13).

This phase of constant velocity ends when all the mass and momentum of CME₁ have been transferred to WS₁, that is, when ${\tau }_{1}={\rm{\Delta }}{\tau }_{1}.$ Applying this condition to Equations (11) and (12), we find the final time,

$$\begin{eqnarray}&&{t}_{f}=\left(\displaystyle \frac{{a}_{1}+\sqrt{{a}_{1}\;{b}_{1}}\;}{{a}_{1}-1}\right){\rm{\Delta }}{\tau }_{1},\end{eqnarray} \tag{ 14 }$$

and the distance,

$$\begin{eqnarray}&&{x}_{{sf}}=\left(\displaystyle \frac{1+\sqrt{{a}_{1}\;{b}_{1}}\;}{{a}_{1}-1}\right){a}_{1}\;{v}_{0}\;{\rm{\Delta }}{\tau }_{1},\end{eqnarray} \tag{ 15 }$$

for this initial, constant velocity phase. After this point, the inner shock disappears and only the ASW feeds the working surface with low velocity material, and the WS₁ enters into a new, decelerated, phase B₁.

2.1.2. Phase B₁

In this second phase, the function F (see Equation (3)) is given by

$$\begin{eqnarray}F & = & {\displaystyle \int }_{{\tau }_{0}}^{0}[{x}_{s}-{v}_{0}\;(t-\tau )]\;\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{0}^{{\rm{\Delta }}{\tau }_{1}}[{x}_{s}-{v}_{1}\;(t-\tau )]\;\dot{{m}_{1}}\;d\tau \\ & = & \displaystyle \frac{1}{2}\;\left\{\dot{{m}_{1}}\;{\rm{\Delta }}{\tau }_{1}\left[2\;{x}_{s}+({\rm{\Delta }}{\tau }_{1}-2\;t)\;{v}_{1}\right]\right.\\ & & -\left.\dot{{m}_{0}}\;{\tau }_{0}\;\left[2\;{x}_{s}+({\tau }_{0}-2\;t)\;{v}_{0}\right]\right\},\end{eqnarray} \tag{ 16 }$$

where the second integral is limited by the duration of CME₁, ${\rm{\Delta }}{\tau }_{1}.$

Using the condition F = 0 and the free-streaming Equation (8) we obtain

$$\begin{eqnarray}&&t=\displaystyle \frac{{\tau }_{0}^{2}-2\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}\;{\tau }_{0}+{a}_{1}\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}^{2}}{2\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}\;({a}_{1}-1)},\end{eqnarray} \tag{ 17 }$$

where the free parameter is now ${\tau }_{0}.$ The position of WS₁ in this second stage (B₁) is obtained by substituting Equation (17) into (8), that is,

$$\begin{eqnarray}&&{x}_{s}=\displaystyle \frac{{\tau }_{0}^{2}+{a}_{1}\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}({\rm{\Delta }}{\tau }_{1}-2\;{\tau }_{0})}{2\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}\;({a}_{1}-1)}\;{v}_{0}.\end{eqnarray} \tag{ 18 }$$

The velocity of WS₁ in this second stage is given by

$$\begin{eqnarray}&&{v}_{s}=\displaystyle \frac{{{dx}}_{s}/d{\tau }_{0}}{{dt}/d{\tau }_{0}}=\displaystyle \frac{{a}_{1}\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}-{\tau }_{0}}{{b}_{1}\;{\rm{\Delta }}{\tau }_{1}-{\tau }_{0}}\;{v}_{0},\end{eqnarray} \tag{ 19 }$$

in terms of the parameter ${\tau }_{0}.$ This velocity steadily decreases from the value at stage A₁ (given by Equation (13)), approaching asymptotically the velocity v₀ of the ASW, for ${\tau }_{0}\to -\infty .$

As expected, Equations (17)–(19) exactly match Equations (11)–(13), respectively, when ${\tau }_{1}={\rm{\Delta }}{\tau }_{1};$ that is, at time t_f and distance x_sf (Equations (14) and (15), respectively).

When CME₁ ends, the ASW is blown out again during an interval of time ${\rm{\Delta }}{\tau }_{0}$ (between $\tau ={\rm{\Delta }}{\tau }_{1}$ and $\tau ={\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1},$ see Figure 2). For simplicity, we have assumed that the parameters of the new ASW are exactly the same than those of the ASW before the first eruption (v₀, ${\rho }_{0},$ and $\dot{{m}_{0}}$). This assumption is not necessary.

At time $\tau ={\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{0},$ a second perturbation (CME₂) is expelled from the Sun with velocity ${v}_{2}={a}_{2}\;{v}_{0},$ mass-loss rate $\dot{{m}_{2}}={b}_{2}\;\dot{{m}_{0}},$ and density ${\rho }_{2}={c}_{2}\;{\rho }_{0}$ (${c}_{2}={b}_{2}/{a}_{2}$) during a finite interval of time ${\rm{\Delta }}{\tau }_{2}$ (see Figure 2). At the time when CME₂ is launched, a new WS₂ is formed, and its dynamical evolution depends on the jump factors a₂ ($\gt 1$), b₂, and c₂, and the duration of CME₂, ${\rm{\Delta }}{\tau }_{2}.$ Its evolution comprises three phases: an initial phase A₂ of constant velocity, a second phase B₂ of acceleration/deceleration, and a third phase C₂ of constant velocity.

2.2.1. Phase A₂

During the initial phase of constant velocity of the trailing WS₂, the function F is given by

$$\begin{eqnarray}F & = & {\displaystyle \int }_{{\tau }_{2}}^{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}[{x}_{s}-{v}_{0}\;(t-\tau )]\;\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}^{\;{\tau }_{3}}[{x}_{s}-{v}_{2}\;(t-\tau )]\;\dot{{m}_{2}}\;d\tau \\ & = & \displaystyle \frac{1}{2}\;\{\dot{{m}_{2}}\;({\tau }_{3}-{\rm{\Delta }}{\tau }_{0}-{\rm{\Delta }}{\tau }_{1})\\ & & \times \left[2\;{x}_{s}+({\tau }_{3}+{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-2\;t)\;{v}_{2}\right]\\ & & -\;\dot{{m}_{0}}\;({\tau }_{2}-{\rm{\Delta }}{\tau }_{0}-{\rm{\Delta }}{\tau }_{1})\\ & & \times \left[2\;{x}_{s}+({\tau }_{2}+{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-2\;t)\;{v}_{0}\right]\}.\end{eqnarray} \tag{ 20 }$$

The result is, of course, the same as that given by Equation (7), but shifted in time by ${\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}.$ Using, again, the free-streaming Equations (5) and (6) in the form

$$\begin{eqnarray}&&{x}_{s}={v}_{0}\;(t-{\tau }_{2}),\end{eqnarray} \tag{ 21 }$$

and

$$\begin{eqnarray}&&{x}_{s}={v}_{2}\;(t-{\tau }_{3}),\end{eqnarray} \tag{ 22 }$$

together with the condition F = 0 in Equation (20), we find,

$$\begin{eqnarray}&&{\tau }_{2}-({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})=-\sqrt{{a}_{2}\;{b}_{2}}\left[{\tau }_{3}-({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})\right],\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&t-({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})=\left(\displaystyle \frac{{a}_{2}+\sqrt{{a}_{2}\;{b}_{2}}}{{a}_{2}-1}\right)\left[{\tau }_{3}-({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})\right],\end{eqnarray} \tag{ 24 }$$

and

$$\begin{eqnarray}&&{x}_{s}=\left(\displaystyle \frac{1+\sqrt{{a}_{2}\;{b}_{2}}}{{a}_{2}-1}\right){a}_{2}\;{v}_{0}\left[{\tau }_{3}-({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})\right],\end{eqnarray} \tag{ 25 }$$

which are equivalent to Equations (10)–(12), but shifted in time by ${\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}.$

Since the material of the ASW and ICME₂ are incorporated into the WS₂ at constant rates, the WS₂ moves with a constant velocity,

$$\begin{eqnarray}&&{v}_{s}=\displaystyle \frac{{{dx}}_{s}/d{\tau }_{3}}{{dt}/d{\tau }_{3}}=\displaystyle \frac{1+\sqrt{{a}_{2}\;{b}_{2}}}{{a}_{2}+\sqrt{{a}_{2}\;{b}_{2}}}\;{a}_{2}\;{v}_{0},\end{eqnarray} \tag{ 26 }$$

which is, of course, similar to Equation (13), but for a₂ and b₂.

This phase of constant velocity lasts until either the ICME₂ or ASW material is completely incorporated into WS₂, and a new accelerated/decelerated phase B₂ begins.

2.2.2. Phase B₂

If the ICME₂ material runs out first, then the WS₂ starts a deceleration phase, because it only sweeps the lower velocity material of the ASW in front of it. On the other hand, if the ASW is incorporated completely into the WS₂ first, then the WS₂ is accelerated by the fast ICME₂ material, which is still being incorporated into the WS₂.

It can be shown that both ICME₂ and the ASW run out at the same time for

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{\tau }_{0}}{{\rm{\Delta }}{\tau }_{2}}=\sqrt{{a}_{2}\;{b}_{2}}.\end{eqnarray} \tag{ 27 }$$

Thus, if ${\rm{\Delta }}{\tau }_{2}/{\rm{\Delta }}{\tau }_{0}\lt \sqrt{{a}_{2}\;{b}_{2}}$, then the ASW runs out first and the WS₂ is accelerated by the incorporation of the fast ICME₂ material, while if ${\rm{\Delta }}{\tau }_{2}/{\rm{\Delta }}{\tau }_{0}\gt \sqrt{{a}_{2}\;{b}_{2}}$, the ICME₂ material runs out first and then the WS₂ is decelerated by the incorporation of the slow ASW in front of it.

2.2.2.1. Phase B₂—Decelerated

Let us first consider the decelerated case (${\rm{\Delta }}{\tau }_{2}/{\rm{\Delta }}{\tau }_{0}\gt \sqrt{{a}_{2}\;{b}_{2}}$). This second phase begins when ${\tau }_{3}={\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{2}$ at a time (Equation (24)),

$$\begin{eqnarray}&&{t}_{i}=\displaystyle \frac{({a}_{2}-1)\;({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})+(\sqrt{{a}_{2}\;{b}_{2}}+{a}_{2})\;{\rm{\Delta }}\;{\tau }_{2}}{{a}_{2}-1},\end{eqnarray} \tag{ 28 }$$

when the working surface is at the distance (Equation (25)),

$$\begin{eqnarray}&&{x}_{i}=\displaystyle \frac{(\sqrt{{a}_{2}\;{b}_{2}}+1)\;{a}_{2}\;{v}_{0}\;{\rm{\Delta }}\;{\tau }_{2}}{{a}_{2}-1}.\end{eqnarray} \tag{ 29 }$$

and ${\tau }_{2i}={\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-\sqrt{{a}_{2}\;{b}_{2}}{\rm{\Delta }}{\tau }_{2},$ from Equation (23).

During this second phase of deceleration the function F is

$$\begin{eqnarray}F & = & {\displaystyle \int }_{{\tau }_{2}}^{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}[{x}_{s}-{v}_{0}\;(t-\tau )]\;\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}^{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{2}}[{x}_{s}-{v}_{2}\;(t-\tau )]\;\dot{{m}_{2}}\;d\tau \;\\ & = & \dot{{m}_{0}}\;\left\{({x}_{s}-{v}_{0}\;t)({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-{\tau }_{2})\right.\\ & & \left.+\displaystyle \frac{1}{2}\;{v}_{0}\left[{({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})}^{2}-{\tau }_{2}^{2}\right]\right\}\\ & & +\;\dot{{m}_{2}}\;\left\{({x}_{s}-{v}_{2}\;t)({\rm{\Delta }}{\tau }_{2}+\displaystyle \frac{1}{2}\;{v}_{2}\;{\rm{\Delta }}{\tau }_{2}\right.\\ & & \left.\times \left[{\rm{\Delta }}{\tau }_{2}+2({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})\right]\right\},\end{eqnarray} \tag{ 30 }$$

where ${\tau }_{2}$ is now the free parameter.

Using the condition F = 0, and the free-streaming equation,

$$\begin{eqnarray}&&{x}_{s}={v}_{0}\;(t-{\tau }_{2}),\end{eqnarray} \tag{ 31 }$$

we find

$$\begin{eqnarray}t & = & \left({({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-{\tau }_{2})}^{2}+{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\;\right.\\ & & \times \left.\{{a}_{2}\left[2\;({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})+{\rm{\Delta }}{\tau }_{2}\right]-2\;{\tau }_{2}\}\right)/\\ & & \left(2\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\;({a}_{2}-1)\right),\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}{x}_{s} & = & (\left({({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-{\tau }_{2})}^{2}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\;\right.\\ & & \times \left.\left[2\;({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-{\tau }_{2})+{\rm{\Delta }}{\tau }_{2}\right]\right)/\\ & & \left(2\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\;({a}_{2}-1))\right)\;{v}_{0},\end{eqnarray} \tag{ 33 }$$

and

$$\begin{eqnarray}{v}_{s} & = & \displaystyle \frac{{{dx}}_{s}/d{\tau }_{2}}{{dt}/d{\tau }_{2}}\\ & = & \displaystyle \frac{({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}-{\tau }_{2})}{({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{b}_{2}\;{\rm{\Delta }}{\tau }_{2}-{\tau }_{2})}{v}_{0},\end{eqnarray} \tag{ 34 }$$

in terms of the free parameter ${\tau }_{2}.$

This phase ends when all of the ASW material has been incorporated into the WS₂, i.e., when ${\tau }_{2}={\rm{\Delta }}{\tau }_{1}$ (see Figure 2). Using this condition, in Equations (32)–(34) we have the time, position, and velocity of the WS₂ at the end of this decelerated phase. They are

$$\begin{eqnarray}{t}_{f} & = & \left({\rm{\Delta }}{\tau }_{0}^{2}+{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\{-2\;{\rm{\Delta }}{\tau }_{1}+{a}_{2}\right.\\ & & \times \left.\left[2\;({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})+{\rm{\Delta }}{\tau }_{2}\right]\}\right)/\\ & & \left(2\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\;({a}_{2}-1)\right),\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{x}_{{sf}}=\displaystyle \frac{{\rm{\Delta }}{\tau }_{0}^{2}+{a}_{2}\;(2\;{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{2})\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}}{2\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\;({a}_{2}-1)}\;{v}_{0},\end{eqnarray} \tag{ 36 }$$

and

$$\begin{eqnarray}&&{v}_{{sf}}=\displaystyle \frac{{\rm{\Delta }}{\tau }_{0}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}}{{\rm{\Delta }}{\tau }_{0}+{b}_{2}\;{\rm{\Delta }}{\tau }_{2}}\;{v}_{0}.\end{eqnarray} \tag{ 37 }$$

Afterward, the WS₂ moves with a constant velocity given by Equation (37) (see Section 2.2.3)

2.2.2.2. Phase B₂—Accelerated

On the other hand, if ${\rm{\Delta }}{\tau }_{2}/{\rm{\Delta }}{\tau }_{0}\lt \sqrt{{a}_{2}\;{b}_{2}},$ the WS₂ is accelerated, since the fast ICME₂ material is incorporated into the WS₂. The acceleration phase of the WS₂ starts at the distance,

$$\begin{eqnarray}&&{x}_{i}=\displaystyle \frac{(1+\sqrt{{a}_{2}\;{b}_{2}})}{\sqrt{{a}_{2}\;{b}_{2}}\;({a}_{2}-1)}\;{a}_{2}\;{v}_{0}\;{\rm{\Delta }}{\tau }_{0},\end{eqnarray} \tag{ 38 }$$

and at a time

$$\begin{eqnarray}&&{t}_{i}={\rm{\Delta }}{\tau }_{1}+\displaystyle \frac{({a}_{2}\;{b}_{2}+\sqrt{{a}_{2}\;{b}_{2}})}{{b}_{2}\;({a}_{2}-1)}\;{\rm{\Delta }}{\tau }_{0},\end{eqnarray} \tag{ 39 }$$

which follow from Equations (23) to (25) with ${\tau }_{2}={\rm{\Delta }}{\tau }_{1}.$

In this phase, the function F is

$$\begin{eqnarray}F & = & {\displaystyle \int }_{{\rm{\Delta }}{\tau }_{1}}^{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}[{x}_{s}-{v}_{0}\;(t-\tau )]\;\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}^{{\tau }_{3}}[{x}_{s}-{v}_{2}\;(t-\tau )]\;\dot{{m}_{2}}\;d\tau \;\\ & = & \displaystyle \frac{1}{2}\{{\rm{\Delta }}{\tau }_{0}\;{\dot{m}}_{0}\;\left[({\rm{\Delta }}{\tau }_{0}+2\;{\rm{\Delta }}{\tau }_{1}){v}_{0}\right.\\ & & +\left.2\;({v}_{2}-{v}_{0})\;t-2\;{v}_{2}\;{\tau }_{3}\right]\\ & & -\dot{{m}_{2}}\;{({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-{\tau }_{3})}^{2}\;{v}_{2}\}.\end{eqnarray} \tag{ 40 }$$

From the condition F = 0 and the free-streaming function,

$$\begin{eqnarray}&&{x}_{s}={v}_{2}\;(t-{\tau }_{3}),\end{eqnarray} \tag{ 41 }$$

we find,

$$\begin{eqnarray}t & = & \left({a}_{2}\;{b}_{2}\;{({\tau }_{3}-{\rm{\Delta }}{\tau }_{1})}^{2}+{\rm{\Delta }}{\tau }_{0}^{2}({a}_{2}\;{b}_{2}-1)\right.\\ & & +\left.2\;{\rm{\Delta }}{\tau }_{0}\;\left[{\rm{\Delta }}{\tau }_{1}\;({a}_{2}\;{b}_{2}-1)-({b}_{2}-1)\;{a}_{2}\;{\tau }_{3}\right]\right)/\\ & & \left(2\;({a}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}\right),\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}{x}_{s} & = & (\left(({a}_{2}\;{b}_{2}-1){\rm{\Delta }}{\tau }_{0}^{2}\;+2\;({a}_{2}\;{b}_{2}-1)\right.\\ & & \times \left.{\rm{\Delta }}{\tau }_{0}\;({\rm{\Delta }}{\tau }_{1}-{\tau }_{3})+{a}_{2}\;{b}_{2}\;{({\rm{\Delta }}{\tau }_{1}-{\tau }_{3})}^{2}\right)/\\ & & \left(2\;({a}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}\right))\;{a}_{2}\;{v}_{0},\end{eqnarray} \tag{ 43 }$$

and

$$\begin{eqnarray}{v}_{s} & = & \displaystyle \frac{{{dx}}_{s}/d{\tau }_{3}}{{dt}/d{\tau }_{3}}\\ & = & \displaystyle \frac{({a}_{2}\;{b}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}+{a}_{2}\;{b}_{2}\;({\rm{\Delta }}{\tau }_{1}-{\tau }_{3})}{({b}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}+{b}_{2}\;({\rm{\Delta }}{\tau }_{1}-{\tau }_{3})}\;{v}_{0},\end{eqnarray} \tag{ 44 }$$

as a function of the free parameter ${\tau }_{3}.$

This phase of acceleration ends when all the momenta of the ICME₂ and the ASW have been transferred to the WS₂, that is, when ${\tau }_{3}={\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{2}.$ This occurs at the time,

$$\begin{eqnarray}{t}_{f} & = & \left((2\;{a}_{2}-1){\rm{\Delta }}{\tau }_{0}^{2}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}^{2}\right.\\ & & +\left.2\;{\rm{\Delta }}{\tau }_{0}\;(({a}_{2}-1)\;{\rm{\Delta }}{\tau }_{1}+{a}_{2}\;{\rm{\Delta }}{\tau }_{2})\right)/\\ & & \left(2\;({a}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}\right),\end{eqnarray} \tag{ 45 }$$

and at the distance from the Sun,

$$\begin{eqnarray}&&{x}_{{sf}}=\displaystyle \frac{({\rm{\Delta }}{\tau }_{0}^{2}+2\;{\rm{\Delta }}{\tau }_{0}\;{\rm{\Delta }}{\tau }_{2}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}^{2})\;{a}_{2}\;{v}_{0}}{2\;({a}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}}.\end{eqnarray} \tag{ 46 }$$

Afterward, the WS₂ moves with constant velocity,

$$\begin{eqnarray}&&{v}_{{sf}}=\displaystyle \frac{{\rm{\Delta }}{\tau }_{0}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}}{{\rm{\Delta }}{\tau }_{0}+{b}_{2}\;{\rm{\Delta }}{\tau }_{2}}\;{v}_{0}.\end{eqnarray} \tag{ 47 }$$

Equations (45)–(47) follow directly from Equations (42) to (44) with ${\tau }_{3}={\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{2}.$

2.2.3. Phase C₂

At the end of the accelerated/decelerated B₂ phase, the WS₂ moves with a velocity given by either Equation (37) or Equation (47), which are the same equation. These equations are the same because they are given by conservation of momentum: in Equation (37) or (47), the numerator represents the sum of the total momenta of the ASW (${\dot{m}}_{0}\;{\rm{\Delta }}{\tau }_{0}\;{v}_{0}$) and the ICME₂ (${\dot{m}}_{2}\;{\rm{\Delta }}{\tau }_{2}\;{v}_{2}$); while the denominator is the sum of their masses (${\dot{m}}_{0}\;{\rm{\Delta }}{\tau }_{0}$) and (${\dot{m}}_{2}\;{\rm{\Delta }}{\tau }_{2}$), respectively. Their ratio is the velocity of the WS₂. This velocity is constant during this phase. Thus, the position of the WS₂ at time t is,

$$\begin{eqnarray}&&{x}_{s}={x}_{{sf}}+{v}_{{sf}}\;\left(t-{t}_{f}\right),\end{eqnarray} \tag{ 48 }$$

where t_f, x_sf, and v_sf are given by the set of Equations (35)–(37) if the WS₂ was decelerated or the set of Equations (45)–(47) if the WS₂ was accelerated during the previous phase B₂.

When the fast WS₂ reaches the slower WS₁, they collide and form a merged region. Due to the conservation of momentum, this merged region travels initially with an intermediate velocity between the velocities that WS₁ and WS₂ had just before the collision.

The time and distance of the collision depends on the initial parameters of the individual events. The collision occurs when all the mass between the two WSs has been completely incorporated. Consequently, the WS₁ is in its decelerated phase B₁ and the WS₂ may be in either: (i) its final phase of constant velocity, (ii) its accelerated phase, or (iii) its decelerated phase. As an example of the formalism of the model, we present below the detailed solution for the first two cases which are related to events described in Section 3.

Case (i) In three of the events (2007 January 24, 2010 May 23 and 2012 November 09), the WSs collide when the WS₁ is in a decelerated phase B₁ while the WS₂ is in phase C₂.

From substitution of the Equations (17) and (18) into the Equation (48), we find the value of ${\tau }_{0}$ at the instant of collision (${\tau }_{0}^{\mathrm{coll}}$). Then, we evaluate the Equations (17) and (18) at ${\tau }_{0}^{\mathrm{coll}}$ to find the time and distance of the collision.

After this point (${\tau }_{0}\lt {\tau }_{0}^{\mathrm{coll}}$), the dynamics of the merged WS is described by the function F,

$$\begin{eqnarray}F & = & {\displaystyle \int }_{{\tau }_{0}}^{0}[{x}_{s}-{v}_{0}\;(t-\tau )]\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{0}^{{\rm{\Delta }}{\tau }_{1}}[{x}_{s}-{v}_{1}(\tau )\;(t-\tau )]\dot{{m}_{1}}\;d\tau \\ & & +{\displaystyle \int }_{{\rm{\Delta }}{\tau }_{1}}^{{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{0}}[{x}_{s}-{v}_{0}\;(t-\tau )]\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{0}}^{{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{2}}[{x}_{s}-{v}_{2}\;(t-\tau )]\dot{{m}_{2}}\;d\tau \\ & = & \;\displaystyle \frac{1}{2}\{{\rm{\Delta }}{\tau }_{0}\;\dot{{m}_{0}}\left[{\rm{\Delta }}{\tau }_{0}+2\;{\rm{\Delta }}{\tau }_{1}-2\;{\tau }_{0}\right]\;{v}_{0}+\dot{{m}_{0}}\;{v}_{0}\;{\rm{\Delta }}{\tau }_{0}^{2}\\ & & +{\rm{\Delta }}{\tau }_{1}\;\dot{{m}_{1}}\left[-2\;{v}_{0}\;{\tau }_{0}+2\;t\;({v}_{0}-{v}_{1})+{\rm{\Delta }}{\tau }_{1}\;{v}_{1}\right]\\ & & +{\rm{\Delta }}{\tau }_{2}\;\dot{{m}_{2}}\left[-2\;{v}_{0}\;{\tau }_{0}+2\;t\;({v}_{0}-{v}_{2})\right]\\ & & +\left[2\;({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1})+{\rm{\Delta }}{\tau }_{2}\right]\;{v}_{2}\}.\end{eqnarray} \tag{ 49 }$$

From the condition F = 0 and the free-streaming equation,

$$\begin{eqnarray}&&{x}_{s}={v}_{0}\;(t-{\tau }_{0}),\end{eqnarray} \tag{ 50 }$$

we find the solution

$$\begin{eqnarray}&&t={c}_{0}+{c}_{1}\;{\tau }_{0}+{c}_{2}\;{\tau }_{0}^{2},\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{x}_{s}={v}_{0}\;\left[{c}_{0}+({c}_{1}-1)\;{\tau }_{0}+{c}_{2}\;{\tau }_{0}^{2}\right],\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{v}_{s}=\displaystyle \frac{{{dx}}_{s}/d{\tau }_{0}}{{dt}/d{\tau }_{0}}=\displaystyle \frac{\left[({c}_{1}-1)+2\;{c}_{2}\;{\tau }_{0}\right]}{({c}_{1}+2\;{c}_{2}{\tau }_{0})}\;{v}_{0},\end{eqnarray} \tag{ 53 }$$

where

$$\begin{eqnarray}{c}_{0} & = & \left({\rm{\Delta }}{\tau }_{0}^{2}+{a}_{1}\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}^{2}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\;(2\;{\rm{\Delta }}{\tau }_{1}\right.\\ & & +\left.{\rm{\Delta }}{\tau }_{2})+2\;{\rm{\Delta }}{\tau }_{0}\;({\rm{\Delta }}{\tau }_{1}+{a}_{2}\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2})\right)/\\ & & \left(2\;\left[({a}_{1}-1)\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}+({a}_{2}-1)\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\right]\right),\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&{c}_{1}=-\displaystyle \frac{{\rm{\Delta }}{\tau }_{0}+{b}_{1}\;{\rm{\Delta }}{\tau }_{1}+{b}_{2}\;{\rm{\Delta }}{\tau }_{2}}{({a}_{1}-1)\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}+({a}_{2}-1)\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}},\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&{c}_{2}=\displaystyle \frac{1}{2\;\left[({a}_{1}-1)\;{b}_{1}{\rm{\Delta }}{\tau }_{1}+({a}_{2}-1)\;{b}_{2}\;{\rm{\Delta }}{\tau }_{2}\right]},\end{eqnarray} \tag{ 56 }$$

in terms of the free parameter ${\tau }_{0}.$

Case (ii): For the event of 2010 August 01 (see Section 3), the collision took place when the WS₁ was in its decelerated phase B₁, while the WS₂ was in its accelerated phase B₂. The time and distance of collision are thus found by solving simultaneously Equations (17) and (18) and Equations (42) and (43). First, we determine the values of ${\tau }_{0}$ and ${\tau }_{3}$ when the collision occurred (${\tau }_{0}^{\mathrm{coll}}$ and ${\tau }_{3}^{\mathrm{coll}}$) and then, we determine the values of t and x_s at that moment. After this point (${\tau }_{0}\lt {\tau }_{0}^{\mathrm{coll}}$ and ${\tau }_{3}^{\mathrm{coll}}\lt {\tau }_{3}\lt {\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{2}$), the further dynamical behavior of the merged WS is described by the function F,

$$\begin{eqnarray}F & = & {\displaystyle \int }_{{\tau }_{0}}^{0}[{x}_{s}-{v}_{0}\;(t-\tau )]\;\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{0}^{{\rm{\Delta }}{\tau }_{1}}[{x}_{s}-{v}_{1}\;(t-\tau )]\;\dot{{m}_{1}}\;d\tau \\ & & +{\displaystyle \int }_{{\rm{\Delta }}{\tau }_{1}}^{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}[{x}_{s}-{v}_{0}\;(t-\tau )]\;\dot{{m}_{0}}\;d\tau \\ & & +{\displaystyle \int }_{{\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}}^{{\tau }_{3}}[{x}_{s}-{v}_{2}\;(t-\tau )]\;\dot{{m}_{2}}\;d\tau \\ & = & {x}_{s}\;\left[{\rm{\Delta }}{\tau }_{0}\;(\dot{{m}_{0}}-\dot{{m}_{2}})+{\rm{\Delta }}{\tau }_{1}\;(\dot{{m}_{1}}-\dot{{m}_{2}})-\dot{{m}_{0}}\;{\tau }_{0}\right.\\ & & +\left.\dot{{m}_{2}}\;{\tau }_{3}\right]+t\;\left[\dot{{m}_{0}}\;{v}_{0}\;({\tau }_{0}-{\rm{\Delta }}{\tau }_{0})-\dot{{m}_{1}}\;{v}_{1}\;{\rm{\Delta }}{\tau }_{1}\right.\\ & & +\left.\dot{{m}_{2}}\;{v}_{2}\;({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-{\tau }_{3})\right]\\ & & +\displaystyle \frac{1}{2}\left[\dot{{m}_{0}}\;{v}_{0}\;({\rm{\Delta }}{\tau }_{0}^{2}+2\;{\rm{\Delta }}{\tau }_{0}\;{\rm{\Delta }}{\tau }_{1}-{\tau }_{0}^{2})\right.\\ & & +\left.\dot{{m}_{1}}\;{v}_{1}\;{\rm{\Delta }}{\tau }_{1}^{2}-\dot{{m}_{2}}\;{v}_{2}\;({\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}-{\tau }_{3})\right].\end{eqnarray} \tag{ 57 }$$

Using the free-streaming equations,

$$\begin{eqnarray}&&{x}_{s}={v}_{0}\;(t-{\tau }_{0}),\end{eqnarray} \tag{ 58 }$$

and

$$\begin{eqnarray}&&{x}_{s}={v}_{2}\;(t-{\tau }_{3}),\end{eqnarray} \tag{ 59 }$$

in Equation (57) we find

$$\begin{eqnarray}&&F={c}_{0}+{c}_{1}\;t+{c}_{2}\;{t}^{2},\end{eqnarray} \tag{ 60 }$$

where

$$\begin{eqnarray}{c}_{0} & = & \displaystyle \frac{1}{2}\;\{{a}_{1}\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}^{2}-({a}_{2}\;{b}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}^{2}\\ & & -2\;{\rm{\Delta }}{\tau }_{0}\;\left[({a}_{2}\;{b}_{2}-1)\;{\rm{\Delta }}{\tau }_{1}-{a}_{2}\;({b}_{2}-1)\;{\tau }_{3}\right]\\ & & -{a}_{2}\;\left[{b}_{2}\;{({\rm{\Delta }}{\tau }_{1}-{\tau }_{3})}^{2}+{\tau }_{3}\;(2\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1}-{a}_{2}\;{\tau }_{3})\right]\},\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}{c}_{1} & = & {\rm{\Delta }}{\tau }_{0}\;({a}_{2}-1)+{a}_{2}\;\left[{b}_{1}\;{\rm{\Delta }}{\tau }_{1}-({a}_{2}-1)\;{\tau }_{3}\right]\\ & & -{a}_{1}\;{b}_{1}\;{\rm{\Delta }}{\tau }_{1},\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&{c}_{2}=\displaystyle \frac{1}{2}\;{\left({a}_{2}-1\right)}^{2},\end{eqnarray} \tag{ 63 }$$

in terms of the free parameter ${\tau }_{3}.$

From the condition F = 0 we then find

$$\begin{eqnarray}&&t=\displaystyle \frac{-{c}_{1}\pm \sqrt{{c}_{1}^{2}-4\;{c}_{2}\;{c}_{0}}}{2\;{c}_{2}},\end{eqnarray} \tag{ 64 }$$

and

$$\begin{eqnarray}&&{x}_{s}={v}_{0}\;(t-{\tau }_{3}),\end{eqnarray} \tag{ 65 }$$

with ${\tau }_{3}$ the free parameter. The sign in Equation (64) is determined by the initial conditions of the collision. During this phase, the velocity of the merged WS can be found from Equation (65),

$$\begin{eqnarray}&&{v}_{s}=\displaystyle \frac{{{dx}}_{s}}{{dt}}={a}_{2}\;{v}_{0}\left[1-\displaystyle \frac{1}{({dt}/d{\tau }_{3})}\right].\end{eqnarray} \tag{ 66 }$$

From Equation (60) and the condition F = 0,

$$\begin{eqnarray}&&\displaystyle \frac{{{dc}}_{0}}{d{\tau }_{3}}+\displaystyle \frac{{{dc}}_{1}}{d{\tau }_{3}}\;t+{c}_{1}\;\displaystyle \frac{{dt}}{d{\tau }_{3}}+2\;{c}_{2}\;t\;\displaystyle \frac{{dt}}{d{\tau }_{3}}=0,\end{eqnarray} \tag{ 67 }$$

thus,

$$\begin{eqnarray}&&\displaystyle \frac{{dt}}{d{\tau }_{3}}=-\displaystyle \frac{{{dc}}_{0}/d{\tau }_{3}+({{dc}}_{1}/d{\tau }_{3})\;t}{{c}_{1}+2\;{c}_{2}\;t}.\end{eqnarray} \tag{ 68 }$$

Substituting Equation (68) in (66) we find:

$$\begin{eqnarray}&&{v}_{s}={a}_{2}\;{v}_{0}\;\left[1+\displaystyle \frac{{c}_{1}+2\;{c}_{2}\;t}{{{dc}}_{0}/d{\tau }_{3}+({{dc}}_{1}/d{\tau }_{3})\;t}\right].\end{eqnarray} \tag{ 69 }$$

Using Equations (61)–(63) in (69), we finally find,

$$\begin{eqnarray}{v}_{s} & = & {a}_{2}\;{v}_{0}\;\left\{1+\left(({a}_{2}-1)\;{\rm{\Delta }}{\tau }_{0}+({a}_{2}-{a}_{1})\right.\right.\\ & & \times \left.{b}_{1}\;{\rm{\Delta }}{\tau }_{1}-({a}_{2}-1)\;{a}_{2}\;{\tau }_{3}+{({a}_{2}-1)}^{2}\;t\right)/\\ & & \left({a}_{2}\;\left[(1-{b}_{2})\;{\rm{\Delta }}{\tau }_{0}+({b}_{1}-{b}_{2})\;{\rm{\Delta }}{\tau }_{1}\right.\right.\\ & & +\left.\left.\left.({b}_{2}-{a}_{2})\;{\tau }_{3}+({a}_{2}-1)\;t\right]\right)\right\},\end{eqnarray} \tag{ 70 }$$

with t given by Equation (64) in terms of ${\tau }_{3}.$

This phase ends when, ${\tau }_{3}={\rm{\Delta }}{\tau }_{0}+{\rm{\Delta }}{\tau }_{1}+{\rm{\Delta }}{\tau }_{2}$ and the merged region enters in a deceleration phase. This deceleration phase is also described by Equations (51)–(53), where now the free parameter is ${\tau }_{0}.$

CME₁ was observed by LASCO on board the SOHO spacecraft (e.g., Brueckner et al. 1995) on 2010 May 23 21:30 UT, with an initial velocity ${v}_{1}=400$ km s⁻¹ and a computed mass of ${m}_{1}=1.5\pm 0.1\times {10}^{16}$ g. The computed CME duration time is ${\rm{\Delta }}{\tau }_{1}=2.85\;\mathrm{hr}.$ Thus, the mass-loss rate is $\dot{{m}_{1}}=3.43\times {10}^{-13}$ ${M}_{\odot }$ yr⁻¹ within a solid angle of ${{\rm{\Omega }}}_{1}/4\pi =0.07$ (where ${{\rm{\Omega }}}_{1}=2\pi [1-\mathrm{cos}({w}_{1}/2)]$ being w₁ the CME₁ width).

The CME₂ was observed by LASCO on 2010 May 24 18:04 UT, therefore the time interval between the CMEs is ${\rm{\Delta }}{\tau }_{0}=17.15\;\mathrm{hr}.$ The initial velocity and computed mass are ${v}_{2}=650$ km s⁻¹ (leading edge) and ${m}_{2}=1.0\pm 0.1\times {10}^{16}$ g, respectively. The computed duration of this CME₂ is ${\rm{\Delta }}{\tau }_{2}=3.71\;\mathrm{hr}$ and $\dot{{m}_{2}}=1.32\times {10}^{-13}$ ${M}_{\odot }$ yr⁻¹ within a solid angle of ${{\rm{\Omega }}}_{2}/4\pi =0.09.$

In Table 1, we present a summary of the parameters reported for this event; the constant factors obtained, by using the parameters in Table 1, are presented in Table 2; and Table 3 shows the comparison of the results of our model and the observations.

Table 1.  Reported Parameters Used in Our Models

Parameter	2007 Jan 24	2010 May 23	2010 Aug 01	2012 Nov 09
${\rm{\Delta }}{\tau }_{0}$ (hr)	14.5	17.15	3.73	10.50
${\rm{\Delta }}{\tau }_{1}$ (hr)	2.0	2.85	1.1	1.55
${\rm{\Delta }}{\tau }_{2}$ (hr)	1.8	3.71	1.0	2.5
m1 (gr)	4.3 × 1015	1.5 × 1016	8.0 × 1015	4.66 × 1015
m2 (gr)	1.6 × 1016	1.0 × 1016	3.0 × 1016	2.27 × 1015
$\dot{{m}_{1}}$ (${M}_{\odot }$ yr−1)	7.21 × 10−14	3.46 × 10−13	2.74 × 10−13	9.04 × 10−14
$\dot{{m}_{2}}$ (${M}_{\odot }$ yr−1)	2.67 × 10−13	1.32 × 10−13	7.39 × 10−13	4.42 × 10−14
w1 (${}^{\circ }$)	85.0	60.0	80.0	90.0
w2 (${}^{\circ }$)	90.0	70.0	100.0	70.0
v0 (km s−1)	310	320	410	300
v1 (km s−1)	600	400	732	500
v2 (km s−1)	1350	650	1138	1100

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Table 3.  Comparison of the Predicted and Observed Values of the ICMEs Interaction Events Indicated in Table 1

MODEL RESULTS
Parameter	2007 Jan 24	2010 May 23	2010 Aug 01	2012 Nov 09
${R}_{\mathrm{coll}}$ (AU)	0.33	0.69	0.20	0.35
${t}_{\mathrm{coll}}$ (hr)	30.8	75.3	13.1	36.4
${t}_{1\;\mathrm{AU}}$ (hr)	71.0	100.2	52.3	95.8
${v}_{1\;\mathrm{AU}}$ (km s−1)	573	427	726	397
OBSERVED DATA
Parameter	2007 Jan 24	2010 May 23	2010 Aug 01	2012 Nov 09
${R}_{\mathrm{coll}}$ (AU)	0.45	0.52–0.77	0.16	0.16–0.46
${t}_{\mathrm{coll}}$ (hr)	30.48	NA	12.91	19–36
${t}_{1\;\mathrm{AU}}$ (hr)	103.6	102.1	60.3	93.2
${v}_{1\;\mathrm{AU}}$ (km s ${}^{-1}$)	716	380	600	350

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The ICME interaction occurred at a distance in the range between 0.55 and 0.8 AU, and the resultant merged region reached the Earth on 2010 May 28 02:00 UT (Lugaz et al. 2012).

We set ${v}_{0}=320$ km s⁻¹, this is the ASW velocity measured in situ by WIND spacecraft on 2010 May 28 01:00 UT, before the arriving of the merged region.

The interaction occurred when WS₁ was in its deceleration phase and WS₂ had already reached its last phase of constant velocity. Figure 3 shows the velocities of ${v}_{s1}$ (panel (a)) and ${v}_{s2}$ (panel (b)) as a function of distance without interaction. The dotted vertical line marks the collision distance.

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By using the Equations (17) and (18) in (29), we predicted that the arrival velocity is ∼427 km s⁻¹, and the in situ arrival velocity was 380 km s⁻¹. Therefore, the difference is 47 km s⁻¹. In terms of time, we obtained a difference shorter than two hours between the predicted and the in situ arrival time. The interaction occurred at a distance of 0.69 AU and a time of 75.3 hr after the CME₁ launch time.

Figure 3 panel (c) shows the trajectories (distance as function of time) of WS₁ and WS₂. The dashed lines represent these trajectories if they did not collide. The trajectory of the merged region is shown as the black thick line. The collision time is marked with the dotted vertical line.

In general, J-maps are used to analyze the collision between CMEs; for comparison with other works we included the elongation (the apparent angular distance computed assuming a fix angle of 90° between the Sun-observer and Sun-ICME direction lines) as function of time in panel (d)).

The event of 2007 January 24 was studied by Lugaz et al. (2008, 2009). CME₁ was observed by LASCO on 2007 January 24 at 16:40:00 UT and CME₂ on 2007 January 25 09:10:00 UT. The event of 2010 August 01 was reported by Temmer et al. (2012). The two eruptions were imaged by LASCO at 04:46:00 UT and 09:35:00 UT on on 2010 August 01, respectively. For the event of 2012 November 09, the most important parameters are reported by Mishra et al. (2014), CME₁ was observed by STEREO A on 2012 November 09 at 20:22:00 UT and the CME₂ on 2012 November 10 at 08:25:00 UT. We refer to these authors for the detailed analysis of each of these interaction events.

As for the event of 2010 May 23, we present in Table 1, a summary of the parameters reported for each event. The change factors (a, b, and c) are presented in Table 2. Table 3 shows the comparison of the results of our model and the observations.

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The events of 2010 May 23 and 2012 November 09 are very well reproduced by our model with differences less than 2 hr (interaction and arrival at 1 AU times) and in velocity less than 50 km s⁻¹. In Figures 3 and 6, respectively, the velocities as functions of distance of WS₁ (panel (a)) and WS₂ (panel (b)) are shown. Both WSs are decelerated due to the fact that the material of the ICMEs are incorporated before the material of the ASW does it. In panel (c), we present the trajectories of the WSs. The collision time is marked as a dotted vertical line. After this time the thick black line shows the trajectory of the merged region that reaches 1 AU. Dashed lines show the trajectories of both WSs if the interaction would not have taken place.

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It is interesting to note that our results for the event of 2010 May 23 do not support the argument of deflection of Lugaz et al. (2012), who found a large deflection of ICME₂ in such way that it never reaches the Earth.

For the event of 2010 August 01, the WS₂ is in phase B₂ at the moment of the interaction (Figure 5). The WS₂ accelerates due to the fact that all of the available ASW material was incorporated to the WS₂ before the ICME₂ material had been engulfed. The difference between the computed and observed interaction times is of a few minutes, while the predicted arrival time differs from the observed value by less than eight hours. This difference may be caused by another CME, which was launched before CME₁ (Liu et al. 2012). This third CME probably interacted with the merged structure (of WS₁ and WS₂) before 1 AU. This interaction may have formed a new merged region which was decelerated in the process, reaching 1 AU in a longer time and with a slower velocity.

The event of 2007 January 24 was in phase C₂ at the moment of interaction (Figure 4). This event has the major differences with the observations due to two factors: (i) the uncertainty on the mass computations: Jackson et al. (2009) reported masses of 2.1 × 10¹⁶ g and 2.0 × 10¹⁶ g for the CMEs whereas Lugaz et al. (2009) reported masses of 4.3 × 10¹⁵ g and 1.6 × 10¹⁶ g, and (ii) the presence of a CIR in the interplanetary medium during its evolution.

The in situ observations show a CIR around the time when the merged region arrival was predicted and in the simulations of this event Lugaz et al. (2008) and Odstrcil & Pizzo (2009)  found that this CIR interacted with the ICMEs. In this case, the model predicted the dynamics of the merged region without the interaction with the CIR. At the present stage, our model is not able to include ICME–CIR interactions. In any case, we have included this event to show that the model gives erroneous outputs when the initial conditions are not set correctly.