RADIAL FLOW PATTERN OF A SLOW CORONAL MASS EJECTION

A great advantage in the interpretation of white-light coronagraph images is the fact that the major part of their brightness signal is produced by Thomson scattering, which depends linearly on the density N_e of the scattering free electrons. The major non-Thomson scattered contributions, like stray light and dust scatter, are removed by subtracting the pre-event background image. The remaining image brightness then should show the CME density distribution produced by Thomson scattering alone. The details of this scattering process have already been developed for more than 50 years and yield for the total image brightness in a given image pixel (Minnaert 1930; van de Hulst 1950; Billings 1966),

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} B & = & {{B}_{\odot }}\frac{\pi {{\sigma }_{e}}}{2}{{\int }_{{\rm LOS}}}F(r,\chi ,u)\ {{N}_{e}}({\boldsymbol{r}} )\ d\ell , \\ F(r,\chi ,u) & = & 2\frac{(1-u)C(r)+uD(r)}{1\;u/3} \\ {} & {} & -\frac{(1-u)A(r)+uB(r)}{1\;u/3}{{{\rm sin} }^{2}}\chi \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where ${{B}_{\odot }}$ is the mean solar brightness of the solar disk, ${{\sigma }_{e}}$ is the coefficient of the differential Thomson scattering cross section, and the parameter u accounts for the solar disk limb darkening. The variables $A,B,C$, and D are known functions (van de Hulst coefficients) of the distance r of the scattering location from the solar center, and χ is the respective scattering angle for a photon from the solar center to the observer. The integration has to be performed along the LOS defined by the effective view direction of the image pixel and s is the geometrical distance along the LOS. We label the different LOSs by ρ, their closest distance to the Sun, and set s = 0 at this closest approach to the Sun so that $r=\sqrt{{{\rho }^{2}}+{{s}^{2}}}$.

Since we do not know the true variation of N_e along the LOS, Equation (1) cannot be evaluated without further assumptions. We adopt the approach proposed by Colaninno & Vourlidas (2009) and Carley et al. (2012) which assumes that the density is concentrated entirely in the mean propagation plane at an angle α of the plane-of-the-sky. Formally, this results in replacing $F(r,\chi ,u)$ in (1) by $F(\rho /{\rm cos} \alpha ,\pi /2-\alpha ,u)$. For the limb darkening coefficient u, we adopt a value of 0.56.

In Figure 3, we show the variation of the weighting coefficient F along a LOS at different distances ρ from the solar center normalized to its value on the propagation plane at angle α. If we knew the shape of the density distribution along the LOS at some distance ρ, the true column density $\int {{N}_{e}}\ d\ell$ differs from the approximate $\int {{\tilde{N}}_{e}}\ d\ell$ derived above by a factor obtained from the convolution of the known normalized density distribution along the LOS with the respective normalized weighting coefficient in Figure 3. The all-in-propagation-plane assumption yields

$$\begin{eqnarray}&&\begin{array}{ccccccccccccccc} {{\int }_{{\rm LOS}}}F(r,\chi ,u)\ {{N}_{e}}({\boldsymbol{r}} )\ d\ell \\ =F(\frac{\rho }{{\rm cos} \alpha },\frac{\pi }{2}-\alpha ,u){{\int }_{{\rm LOS}}}{{\tilde{N}}_{e}}({\boldsymbol{r}} )\ d\ell \\ \;{\rm or}\;\quad \frac{{{\int }_{{\rm LOS}}}{{N}_{e}}({\boldsymbol{r}} )\ d\ell }{{{\int }_{{\rm LOS}}}{{\tilde{N}}_{e}}({\boldsymbol{r}} )\ d\ell } \\ ={{[{{\int }_{{\rm LOS}}}\frac{F(r,\chi ,u)}{F(\frac{\rho }{{\rm cos} \alpha },\frac{\pi }{2}-\alpha ,u)}\ \frac{{{N}_{e}}({\boldsymbol{r}} )}{{{\int }_{{\rm LOS}}}{{N}_{e}}({\boldsymbol{r}} )\ d\ell ^{\prime} }\ d\ell ]}^{-1}}. \\ \end{array}\end{eqnarray} \tag{ 2 }$$

We note that the normalized weighting coefficient hardly depends on the distance ρ of the LOS from the solar center. That is, a CME plasma volume propagating radially with a constant angle of propagation off the POS, i.e., with a constant $s/\rho$, receives the same LOS weighting. For an extended CME with an unknown LOS distribution, the correction we have to apply to the column density determined by the all-in-propagation-plane assumption therefore hardly depends on ρ if the CME propagates strictly radially.

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We further note that a closer inspection shows that the s dependence of the weighting coefficient in Figure 3 is due to the approximate $1/{{r}^{2}}$ dependence of the van de Hulst coefficients which reflects the decrease of the incident Sun light with distance from the solar center. The influence of the variation with scattering angle through the differential Thomson cross section is only minor.

After the conversion of the excess brightness into the column electron density per pixel in the CME region, we can estimate the according column mass assuming a He²⁺/H⁺ composition of 10%. To analyze the mass transport inside the CME, we integrate the CME column mass density from each pixel over discrete shells of radius r centered on the Solar center to obtain the mass $m(r,t)$ per unit length in radial direction at the times t of the respective images. The discretization of the radial shells is indicated in the bottom right panel of Figure 1. To define the shells, we have divided the radial range from 1.6 to 14.8 R_S into 39 equal intervals, each 0.34 R_S wide. This discretization is large enough to raise the integrated mass in each shell significantly above the noise level, and small enough to sufficiently resolve the radial mass density distribution. In order to reduce the noise in the estimates of m, we limit the integration in latitude to the smallest possible sector which includes the CME. Note that we also excluded the narrow wedge near the center of the CME where the pre-event streamer disturbs the excess brightness.

There are two possible sources of systematic error in $m(r,t)$. One error in $m(r,t)$ may be due to the assumption in Equation (2) about the LOS distribution of the CME mass, in particular, due to a wrong longitudinal propagational angle α. A second error in $m(r,t)$ may have been introduced by the somewhat arbitrary selection of the latitudinal integration boundaries. Choosing them to be too wide causes additional noise to be added to $m(r,t)$. To estimate these possible errors, we repeated the calculation of $m(r,t)$ 12 times for slightly different integration boundaries and also varied the propagation angle α within $22\pm 5{}^\circ$. As an example, in Figure 4, the average and $3\sigma$ uncertainty of the radial mass density profile from its 12 integrations at 21:39 UT is displayed in green.

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Figure 5 displays the whole time series of the mass density profiles $m(r,t)$ thus obtained. The colors from black to red indicate time t in increasing order. On each curve, the asterisk marks the position of the hand-traced leading edge. We can see that the leading edge corresponds to finite, different mass densities which slightly increase with distance. There is still some residual CME mass ahead of the leading edge. The spikes in $m(r,t)$ at about $r\gt 10\;{{R}_{S}}$ are due to the increasing noise in the coronagraph images toward the outer boundary of the field of view.

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In order to derive other quantities from $m(r,t)$ and their uncertainty, we first fitted the smoothing splines obtained thus far. The amount of smoothing was controlled by the variance of the individual profiles from their average. One such smoothing spline is shown in black in Figure 4. Note that well ahead of the CME, $m(r,t)$ was set to zero before the smoothing. The mass measurements start to be sufficiently reliable at $\simeq 2\;{{R}_{S}}$, about 0.5 R_S above the COR1 occulter.

If we assume that there is no mass contribution from the pile up of the solar wind around the CME, then the mass flow is described by a continuity equation,

$$\begin{eqnarray}&&\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho {\boldsymbol{v}} )=0.\end{eqnarray} \tag{ 3 }$$

The radial mass density profiles $m(r,t)$ that we have derived above are integrals of the volume mass density ρ over cylinder surface sections which are spanned by the weighted LOS integration in one direction and the heliographic latitude section in the other. Clearly, if ${\boldsymbol{v}}$ has only a radial component, then the $\nabla \cdot$-operation commutes with both integrations and Equation (3) could be transformed to

$$\begin{eqnarray}&&\frac{\partial m}{\partial t}+\frac{\partial }{\partial r}(m{{v}_{r}})=0.\end{eqnarray} \tag{ 4 }$$

We expect that the major velocity component of the CME is indeed radial so that Equation (4) provides a good approximation of the evolution of the radial mass density m.

It is obvious that the mass flow in the latitudinal direction can be accounted for by partial integration and does not change Equation (4) as long as the longitudinal integration boundaries were chosen wide enough to include the entire visible CME signal. The effect of a latitudinal component of the mass flow is more difficult to estimate. From Figure 3, we immediately see that a change in the propagation angle α or the LOS density profile in the CME with distance ρ could lead to a brightness change without changing the true column mass density. However, from many CME observations, it has become evident that the non-radial components of the CME propagation are small. In the following, we will therefore drop the index r of v_r in Equation (4) and assume this equation to be an adequate description for the evolution of m, neglecting possible changes in the mass density caused by the non-radial components of ${\boldsymbol{v}}$.

At first sight, Equation (4) appears to be of little value as long as $v(r,t)$ is not known. However, since $m(r,t)$ is measured, it provides us with a chance to derive the velocity. Given that $m(r\to \infty ,t)$ vanishes, an integration of Equation (4) over the radius with variable lower bound yields

$$\begin{eqnarray}&&v(r,t)={\mkern 1mu} \frac{1}{m(r,t)}\ \frac{\partial }{\partial t}\int _{r}^{\infty }m(r^{\prime} ,t)dr^{\prime} .\end{eqnarray} \tag{ 5 }$$

That is to say, at a given time, the total mass change beyond r comes from the mass flow through the shell at r.

In Equation (5), the calculation of the velocity requires the time derivatives of mass profiles. To reliably calculate these derivatives, we interpolated the mass profiles using cubic splines. The result of such interpolations is presented in Figure 6. The first mass profile at 19:39 UT is displayed in black, and the following mass profiles change color with time from violet to red. The statistical error in $v(r,t)$ was obtained by repeating the calculation 300 times with mass density profiles $m(r,t)$ contaminated by random noise profiles of an amplitude which agrees with the standard deviation determined for $m(r,t)$. The variation in the resulting velocities was used to determine the error in $v(r,t)$. In such a way, we can derive a Eularian flow field within the CME at each interpolated spacetime position. Some of the velocity profiles and associated error bars at defined shell positions and at selected representative times are displayed in Figure 7. The solid curve in each panel is a quadratic fit to the derived velocity profile. Figure 8 shows an example of the time evolution of the Eulerian velocity at a given distance. We find that the Eulerian velocity decreases with time at all distances.

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So far, we have ignored possible mass sources in the CME continuum Equations (3) and (4). One such possibility which has been discussed is the pile up of solar wind mass in front of a fast propagating CME. We see similar density enhancements ahead of fast solar wind streams where they run into the warped up current sheet. Formally, this could be taken into account by adding a source term in Equation (4). The effect on the velocity derived can be seen straight forwardly. If a source is present, then

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} \frac{\partial }{\partial t}m & = & -\frac{\partial }{\partial r}(vm)+s(r,t) \\ {\rm Set}\qquad f(r,t) & = & {{\int }^{r}}s(r^{\prime} ,t)dr^{\prime} \quad {\rm then} \\ \frac{\partial }{\partial t}m & = & -\frac{\partial }{\partial r}(vm)+\frac{\partial }{\partial r}f=-\frac{\partial }{\partial r}(({\underbrace{v-f/m}_{{{v}^{\prime }}}})m). \\ \end{array}\end{eqnarray*}$$

Hence, the source term is absorbed in the flow velocity which we obtain from Equation (5). For example, if there is mass pile up in front of the CME, then f should grow near the front side of the CME from zero to the piled-up mass well ahead of the CME. This should lead to a strong decrease of the velocity v' since, at the front side of the CME, m drops with distance and the velocity contamination f/m increases even stronger than f alone. Therefore, mass pile up should appear as a decrease in the velocity profile near the CME front. In Figure 7, we see that the velocity at the CME front is consistently about 450–500 km s⁻¹. Note that the error in the most distant velocity estimate has a relatively large uncertainty because it was obtained in Equation (5) from a division by a small mass density $m(r,t)$.

To check the consistency of the velocity $v(r,t)$ derived above, we have used it in the continuity equation, i.e., Equation (4), to predict the radial mass density profile $m(r,{{t}_{j}})$ from a given initial value $m(r,{{t}_{j-1}})$. Both times ${{t}_{j-1}}$ and t_j were chosen to coincide with the observational times the coronagraph images were taken so that the respective mass density profiles were those obtained directly from the integration of the image data. The continuity Equation was integrated by a straight-forward Lax–Wendroff scheme. The boundary condition varies linearly from $m({{r}_{0}},{{t}_{j-1}})$ to $m({{r}_{0}},{{t}_{j}})$ at each time step.

Figure 9 shows examples at two pairs, ${{t}_{j-1}}$ and t_j, of successive image times. The solid black and orange lines are the measured mass density profiles at ${{t}_{j-1}}$ and t_j, respectively. The dashed green line is the profile calculated from ${{t}_{j-1}}$ to t_j using the flow speeds $v(r,t)$ derived in the previous subsection. It can be seen that the predicted and measured mass density profiles fit fairly well.

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A change of Equation (4) to the Lagrangian frame is achieved first by integrating the velocity to the Lagrangian or material path of a mass element:

$$\begin{eqnarray}&&\begin{array}{ccccccccccccccc} \frac{\partial }{\partial t}R(t,{{t}_{0}})=v(r=R(t,{{t}_{0}}),t) \\ \quad {\rm with}\;{\rm initial}\;{\rm condition}\quad R({{t}_{0}},{{t}_{0}})={{r}_{0}}. \\ \end{array}\end{eqnarray} \tag{ 6 }$$

In this formulation, the paths R are labeled according to the time t₀ (2nd argument) when they pass the distance ${{r}_{0}}\simeq 2\;{{R}_{\odot }}$ where we start our velocity derivation.

Along these paths, the advected mass density $m(r=R(t,{{t}_{0}}),t)$ then changes in time according to

$$\begin{eqnarray}&&\frac{dm}{dt}=-m\frac{\partial v}{\partial r}(r=R(t,{{t}_{0}}),t).\end{eqnarray} \tag{ 7 }$$

The error bars assigned to each path represent the estimated uncertainty obtained again by repeated integrations of Equation (6) of the average flow speed perturbed with random noise profiles of amplitude in agreement with the standard deviation of $v(r,t)$.

From the derived Eularian velocity field in the last section, it is possible to derive the Lagrangian path of a mass element starting from r₀ at time t₀. In Figure 10, we present the Lagrangian path of 10 mass elements starting at 19:39 UT. At 19:39 UT, the CME mass covers a range from 2 to 3.8 ${{R}_{\odot }}$. The selection of 10 mass elements is performed to make the error bars of the trajectories distinguishable between neighboring trajectories, and at the same time to sufficiently resolve the properties of different trajectories. For the CME studied here, each mass element propagates at almost a constant Lagrangian velocity. The height–time diagram of the Lagrangian paths in Figure 10 closely resembles a fan; however, each Lagrangian path (and those in between) carries a different amount of mass density. If we approximate these velocities by exact constants, it is easily seen that $\partial v/\partial r\propto 1/t$ and the mass density along a Lagrangian path decrease as $\propto 1/R$.

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Any significant derivation of the Lagrangian path from a straight line indicates either a physical acceleration $dv/dt$ of the CME masses or, as discussed in Section 3.3, the presence of a mass source. In the example studied here, we conclude from the constancy of the Lagrangian trajectories that there is may be no evidence for either of the two.

We note that the asterisks, which represent the radial advance of the leading edge, are attached to the fastest Lagrangian path in the beginning but systematically fall slightly behind at later times. The leading edge we have picked by visual inspection (see Figure 2) may therefore not correspond exactly to a material path as the name suggests. It seems that as the CME front dims away at increasing distance from the Sun, there is a tendency to place the leading edge position further inward where the image brightness is more enhanced.

Another interesting aspect of Figure 10 is the possibility of extrapolating the Lagrangian paths backwards. We find that the fastest path with a mean velocity of 521 km s⁻¹ intersects the solar surface at 18:35 UT, about one hour after a flare in the EUV images indicates the initiation of the eruption at ${{t}_{{\rm flare}}}=17:30$ UT. The slower velocity paths extrapolate to the surface at increasingly later times, while the slowest velocity path with a mean of 283 km s⁻¹ intersects r = 1 at 18:58 UT. A mass element experiencing approximately a constant acceleration between some time t_start and t_end will propagate afterwards on a Lagrangian path which intersects the solar surface at $({{t}_{{\rm start}}}+{{t}_{{\rm end}}})/2$, irrespective of the strength of the acceleration. If this simple model holds true, then the fast masses (red trajectories in Figure 10) must have been accelerated earlier than the slow masses (dark blue trajectories).

Usually, the kinetic energy of a CME is estimated by simply using its total mass and leading edge speed. This ignores the extended distribution of CME mass and velocity. For the CME we analyzed here, we see systematically lower speeds with increasing distance from the leading edge. Therefore, the conventional approach probably tends to overestimate the kinetic energy of a CME.

In the upper panel of Figure 11, we show the kinetic energy of the CME between successive shells at r_i and ${{r}_{i+1}}$ at a few select times

$$\begin{eqnarray}&&d{{E}_{{\rm k}}}({{r}_{i}},{{r}_{i+1}},t)=\int _{{{r}_{i}}}^{{{r}_{i+1}}}\frac{m(r,t)}{2}{{v}^{2}}(r,t)\ dr.\end{eqnarray} \tag{ 8 }$$

The non-uniformity of the kinetic energy of CME shells can be clearly seen. In Figure 12, the time evolution of the total kinetic energies of the entire CME E_k are marked in black. The values derived with the conventional method indicated by the dashed line can be four times larger than the values obtained with our more refined method, indicated by the solid line. This large discrepancy mainly comes from the v² term in Equation (8).

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Another energy we estimated is the gravitational potential energy above the solar surface according to

$$\begin{eqnarray}&&d{{E}_{{\rm p}}}({{r}_{i}},{{r}_{i+1}},t)=\int _{{{r}_{i}}}^{{{r}_{i+1}}}m(r,t){{g}_{\odot }}\frac{{{R}_{S}}}{r}(r-{{R}_{S}})\ dr.\end{eqnarray} \tag{ 9 }$$

The potential energy of each shell at the same selected times as the kinetic energy is illustrated in the lower panel of Figure 11. The non-uniformity of the potential energy is also evident. The total potential energy E_p plotted in red in Figure 12 reveals that E_p estimated using the total mass and leading edge distances may also overestimate the true value. The discrepancy between E_p derived with these two different methods is much smaller compared with that of E_k.