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\title{The April 03, 2010 CME and its Corresponding Driven Shock Wave}
\author{Phillip Hess}

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\begin{abstract}

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\section{Introduction}

Coronal Mass Ejections (CMEs) and their counterpart at the Earth, Interplanetary Coronal Mass Ejections (ICMEs) are a major driver of space weather effects at the Earth. A CME is an eruption of magnetized plasma from the solar. Extreme cases of CMEs show speeds over 3000 km/s and carry as much as 1032 ergs of energy (Howard et al. 1985, Hunhausen, 1987, 1999). 

An ICME at the Earth will cause geomagnetic activity if its magnetic field is mostly in the southern direction, making it oppositely aligned to the field of the Earth. This will cause magnetic reconnection along the magnetopause, opening the magnetospehere and allowing solar wind energy inside. These geomagentic storms can damage a wide array of technological systems, including satellites, power grids and GPS navigation systems as well as endangering astronauts in space. Therefore, being able to understand the processes governing CME evolution and predicting an arrival at the Earth are crucial to our increasingly technological society.

Before the launch of the Solar Terrestrial Earth Observatory (STEREO) Mission in 2006, attempts to study CME evolution were limited to just two sets of data, near Sun white light observations from coronographs and in-situ solar wind observations near the Earth. The space in the heliosphere between these two regions was completely unobserved, making understanding of the evolution of CMEs difficult. However, STEREO changed this with the Sun Earth Connection Coronal and Heliosperic Investigation (SECCHI) suite of instruments, which provide a continuous view from the Sun to the Earth. Also, the STEREO mission is composed of two nearly identical spacecraft in near Earth orbits around the Sun. This allows for stereoscopic imaging of the Sun. This, combined with SECCHI, allows  the 3-dimensional tracking of CMEs and their shocks all the way from the Sun to the Earth.

Many works have already been performed using SECCHI data to study the velocity of CMEs in the heliosphere \ref{(Wood & Howard 2009; Lugaz et al. 2009; Rouillard et al. 2009; Howard & Tappin 2009; Moestl et al. 2009; Liu 2010)}. Most of these studies were performed by taking the position of the leading edge of the CME at different times to calculate a velocity. However, very few works have applied this same technique to the other front that can be associated with the CME, its driven shock front.

There are two different types of CMEs, fast and slow, that are differentiated by their initial velocities near the Sun. A slow CME will be slowly accelerated to the speed of the solar wind before the velocity tapers off and becomes approximately constant. Fast CMEs are much the same, as they are decelerated to the solar wind velocity before reaching a constant speed. 

In addition to the ICME at the Earth, geomagnetic storms can be caused by CME driven shock waves. A CME that is initially traveling much faster than ambient solar wind will produce a shock wave, which can open the magnetosphere with a southward magnetic field and can also accelerate energetic particles that are known to have a negative impact on satellites. Therefore, just as understanding CME propagation is important to space weather, understanding the evolution of the shock is just as important. Fortunately, it has been shown before that in addition to having obvious structures in in-situ solar wind data, shocks of sufficient strength can also be seen in white light images \ref{(Maloney & Gallagher 2011; Bemporad & Mancuso 2010; Ontiveros & Vourlidas 2009)}. It is then possible to use SECCHI to track both the CME and the shock it drives though the interplanetary space and relate it to white light signatures at the Earth.

To clear up confusion in the rest of the paper, the following are the terminology will be used to describe CME and ICME structure throughout the rest of this paper:

	The ejecta front is the CME material that is actually ejected from the corona. The term flux rope  will be used almost interchangeably, as the ejecta front is assumed to be a twisted magnetic flux tube structure during its propagation. In the classical 'three part' CME structure (CITE), this area is considered to be the cavity. The ejecta front is generally low in density, giving it a darker appearance in coronograph and heliospheric imager data. 

	The shock front is the outermost front seen in images. It is a shock wave caused by a very fast CME as it travels through the interplanetary space. In images it is seen as a usually sharp jump from a bright, dense region to the darker ambient solar wind material.

	In between these two fronts, the sheath region is an area of accumulated solar wind material in front of the ejecta that is highly dense and therefore the brightest part of the CME structure as seen in the images.

	ICME will hereafter only refer to CME signatures as seen in-stiu with the ACE spacecraft. The different signatures that will be discussed are the shock, which is seen as a sharp increase in magnetic field strength, velocity and density (CITE); the sheath, a high density region between the shock and the flux rope; and the classical flux rope signature, a rotating magnetic field (meaning that in the north-south direction the magnetic field direction goes from either positive to negative or negative to positive), a temperature that is significantly lower than expected based on the density, and a region where magnetic pressure is significantly larger than thermal pressure, which can also be considered a region of very low $\beta$.

\section{Methods}

Despite the advantages in tracking Coronal Mass Ejections provided by STEREO, it is still far from a trivial task. Many different methods and models using STEREO data have been developed (INCLUDE CITATION). For the purpose of studying the evolution of the shock and CME ejecta fronts, two different methods were used in this study.

The first of these methods is a forward modeling technique which uses the Graduated Cylindrical Shell, or croissant model, using the Raytrace code developed by Arnaud Thernisien (CITATION) This model involves taking a cylindrical flux rope shape and superimposing it upon images from multiple spacecraft at once, using six free parameters to best match the model to the images. These parameters include the direction of propagation (longitude and latitude), the leading edge distance of the ejecta from the Sun, the half angle width of the shell, the tilt angle of the central axis orientation, and the ratio between the major and minor radius of the flux rope (aspect ratio). Making the assumption that the other five parameters are roughly constant as the CME propagates, the leading edge height can be used to determine a height profile of the CME with time. Performing a numerical derivative of this data can then yield a velocity. In addition to fitting the flux rope, a simplified model for measuring the shock height is also used. While the model does not match the shock nearly as well as the flux rope, it is still useful for providing a leading edge height. This height can be compared to the flux rope height to provide information about the sheath region in between the two fronts.

After deciding upon this model, there was still the issue of how to properly apply it to study the two distinct fronts of the CME. STEREO coronographs and heliospheric imagers observe white light scattered by electrons. So, the brightness of an area in an image is proportional to the density of that region. Flux ropes are generally known to be of lower density than the ambient solar wind (CITATION), while the regions in front of the flux rope that comprises the sheath is solar wind material that gets piled up in front of the flux rope and should therefore be of a higher density. Also, the basic shock jump conditions would indicate that the shock would be along the region where there is a sharp increase in density. Therefore, what can be expected in images is a fairly dark ambient solar wind background, with many brightness enhancements indicating the sheath region surrounding a dark core that is the flux rope. (THROW SOME CITATIONS IN HERE) Therefore, using the GCS model, the core is generally fit as the flux rope ejecta front and the shock front is considered to be the very front of the bright region seen in images.

While others have been successful using the Raytrace model to track CMEs (CITE POSSIBLE), another method was sought as validation for the Raytrace measurements. Using so-called jmaps (CITE SHEELY), which are stack plots along a given position angle from one spacecraft that provide plots in terms of elongation angle vs. time, other height measurements were also taken. However, due to various observational effects \ref{(Howard 2009, Rollet, Lugaz)} getting heights in distance from the Sun from jmaps is not as easy as just converting the elongation angles to a distance. There are many different methods for deriving true distances. In this work, two were used.

Close to the Sun, primarily in the STEREO COR2 field of view (4-15 Rs) the fixed-$\varphi$ method was used. This method gives distances based on the equation (PROVIDE FORMULA) where $\varepsilon$ is the elongation angle of the object being tracked, as determined from the jmap and $\varphi$ is the direction of propagation of the CME, relative to the observer. While most work in the past that has been done has attempted to fit the data to provide the best direction of propagation, this study is different in that the propagation direction used to fit the data in the GCS model was used, allowing for a straight-forward calculation of leading edge distances.

Near the Sun, fixed-$\varphi$ is fairly accurate. However, as distances increase to be further out into the heliosphere, this method introduces errors, over-estimating the distance of the leading front of an object. (For detailed discussion of this, see Rollet) For this reason, another method of extracting distances from the jmap data discussed in the same paper was used for data further from the Sun, the harmonic mean method given by (PROVIDE FORMULA) This method, while similar to the fixed-$\varphi$ method, includes a geometric correction to account for the over-estimation. Combing the two methods near and further away from the Sun, one data set could be created for judging the CME height profile based on observations from just one spacecraft.

While these methods solved the problems of how to convert measurements from jmap elongation angles into heights, there was still the issue of how to interpret the jmap data for the purposes of studying the separate fronts of the CME. Since a jmap is built using the same white light images that were used in the Raytrace fitting, the same basic logic applies. The jmap figures actually show a bright streak denoting the CME, so it was concluded that the bright feature observed is the high density sheath region, with the outermost front being the shock front and the inner front being the ejecta front. 

While this general assumption of the fronts should be valid, there is one very significant possible issue that should be considered with this interpretation of jmap data. Jmaps are constructed using running difference images, meaning the previous image to the one currently being used is subtracted out to highlight the new features that emerge. In the case of the shock front, this should not matter much, as the shock front is generally traveling into an empty region of space and can be well seen in a running difference image. However, if the sheath region is larger than distance traveled by the shock front between two images, the shock region that gets subtracted out from the previous image will remove the   back of the sheath and will leave a dark void. In a jmap it is possible that this void would be interpreted  as the the ejecta when it is actually an observational effect. This, in conjunction with the general errors of smoothing the data needed to construct a coherent jmap should be considered when handling this data set.

\section{Results}

Applying these methodologies to the April 03, 2010 CME, both the shock front and ejecta front were tracked. Extreme Ultraviolet Images and GOES X-Ray data show an eruption occurring around 10:00 UTC on April 03. The eruption was associated with a relatively small flare and occurred from NOAA active region 11059. The CME was well observed in both STEREO A and B, as well SOHO LASCO, in the low corona. The CME could be clearly seen in STEREO A all the way to the Earth, into the STEREO HI-2 field of view. It was not as clear in STEREO B HI-2 due to the presence of the Milky Way in the background. 

The ICME was also observed in-situ by ACE. The shock is seen as a spike in the Magnetic Field, Velocity and Density at 08:00 UTC on April 05, 2010. The sheath lasts for three and and half hours until 11:30 UTC and initially posses a very strongly southern magnetic field which is very likely the cause of a small geomagnetic storm with a dst peak of -77 nT at 15:00 UTC on April 06. The ejecta front, starts at 11:30 UTC on April 05 and the ejecta as a whole lasts until 15:00 UTC on April 06. The initial magnetic field is pointed northward and rotates southward, while also presenting a low $\beta$ and a low temperature. 

The height profiles from the GCS method and the jmap data obtained from this event are all shown in (FIGURE) out to a distance of around 60 RSun. Each of the three independent data sets shows a strong agreement in the the height profiles for both fronts, with the shock front in particular showing a strong correlation in each data set. It is not surprising that the shock front data shows a better agreement than the ejecta front data due to the previously mentioned errors in measuring the ejecta front in the jmap data.

The velocities for the CME for each data set, calculated with a numerical derivative, is shown in (FIGURE). These velocities show a bit more variation from data set to data set than the height data, but many similarities can be observed. It is obvious that the CME reaches a maximum speed of well over 1000 km/s before declining to a speed between 800 and 900 km/s. These values make sense, as the total time between the eruption and the arrival of the ejecta at ACE is about 49.5 hours, yielding an average velocity of 833 km/s. 

In order to determine some of the basic parameters of the CME velocity profiles from each data set, the data was fit with an exponential of the form (INCLUDE FORMULA). Where v is the velocity at a given time, vi is the initial velocity of the CME, vf is the final velocity of the CME and tc is the e-folding time it takes for the CME to reach its final asymptotic velocity. The fitting parameters from each method are given in (TABLE) 

(FIGURE) shows the standoff distance, or the distance between the leading edge of the shock front and the leading edge of the ejecta front from the GCS data set. This data set was chosen because it was considered the most reliable and accurate. For the first 60 solar radii at least, the increase in standoff distance increases in an apparently linear fashion. To see if this linear evolution was possible, the standoff distance was fit with a line that was extrapolated out to 1 AU and compared to the observed sheath size at ACE, as calculated by taking the duration and velocity of the sheath. The extrapolated linear sheath would have been NUMBER RSun, which is much larger than the calculated observation of NUMBER RSun. If the calculated standoff distance within 60 RSun is believed, it cannot continue to increase at the same rate. An exponential fit, being applied to the same data would match this leveling off of the standoff distance. The exponential fit yielded a NUMBER  RSun standoff distance, much closer to the in-situ data. It should also be noted that comparing the standoff distance as measured from STEREO and in-situ data is not a direct 1:1 comparison, as the Raytrace measurements are along the CME leading edge and the in-situ data is of whatever part of the CME strikes the Earth.

\section{Discussion and Conclusions}

While too much cannot be determined from one single event, the results from the April 03 event did provide several encouraging results. The strong correlation, especially as it comes to shock fronts, for both the Raytrace and jmap methods from each satellite indicates that the methods are reasonably accurate for carrying out CME measurements. 

In addition to the agreement between different methods of measurement, the results also make sense because the velocity profiles produces match with known theories of CMEs. The velocity profiles from each show the CME reaching a high initial speed before decreasing to a constant speed. This is also reinforced by the ACE data, which shows a comparable velocity to the constant velocity in the plots. INCLUDE NUMBERS. 

Another test of the data that was performed was an extrapolation of the fits to 1 AU, in order to generate an arrival time that could be compared to the observed arrival at ACE. However, as previously mentioned, the data being used is not measuring the same part of the CME as ACE. The propagation direction of the CME as measured in Raytrace was about 27 deg from the Sun-Earth line. It is difficult to know how much of a difference this will cause, because even using the assumed GCS geometry, CMEs can have their shapes significantly distorted by accreted solar wind mass in front of the flux rope, and this could also have an impact on the shape of the shock in front of the flux rope. (Howard 2012)

It is still worth comparing the predicted arrival times to the ACE data, as even with the aforementioned errors the prediction should be close, even with this simple empirical model. (FIGURE) shows different predictions, based on different subsets of the data from both the Raytrace measured data and the data from jmap A all the way out to the Earth. The jmap B data was not used, since the HI-2 data does not clearly show the event. This is to see if accurate predictions are possible using only a small amount of data near the Sun. The predictions were generated by taking fits on the velocity data and integrating the functions do to get a function of position vs. time. These functions were then solved in order to determine the time of the CME at 1 AU. 

The different arrival times predicted for the shock especially are all encouraging, but all are within a few hours, even using data just a quarter of the way from the Sun to the Earth. However, since it is just one event the accuracy of this prediction method is still far from confirmed. It is encouraging though that this simple empirical formula was able to generate such accurate predictions for any event. In the future, more events must be studied in order to test this method further.

In general, the methods used in this work show promise, but before any definitive conclusions can be drawn, more events must be studied in great detail. Future plans call for a catalog of events that display clear shock signatures at the Earth and can be studied in the same was as the April 03, 2010 event to provide a validation of the methods. Finding CMEs that work as well as the April 03 CME will not be trivial, as this CME was both directed very near the Earth and propagated in the interplanetary space free of any noticeable interaction with any other interplanetary transients.





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