Inconsistencies Between Local and Global Measures of CME Radial Expansion as Revealed by Spacecraft Conjunctions
N. Lugaz, T. M. Salman, R. M. Winslow, N. Al-Haddad, C. J. Farrugia, B. Zhuang, A. B. Galvin
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Inconsistencies Between Local and Global Measures of CME Radial Expansionas Revealed by Spacecraft Conjunctions
No´e Lugaz, Tarik M. Salman, R´eka M. Winslow, Nada Al-Haddad, Charles J. Farrugia, Bin Zhuang, andAntoinette B. Galvin Space Science Center and Department of Physics and Astronomy, University of New Hampshire8 College RdDurham, NH, 03824, USA
Submitted to ApJABSTRACTThe radial expansion of coronal mass ejections (CMEs) is known to occur from remote observations;from the variation of their properties with radial distance; and from local in situ plasma measurementsshowing a decreasing speed profile throughout the magnetic ejecta (ME). However, little is known onhow local measurements compare to global measurements of expansion. Here, we present results fromthe analysis of 42 CMEs measured in the inner heliosphere by two spacecraft in radial conjunction.Themagnetic field decrease with distance provides a measure of their global expansion. Near 1 au, thedecrease in their bulk speed provides a measure of their local expansion. We find that these twomeasures have little relation with each other. We also investigate the relation between characteristicsof CME expansion and CME properties. We find that the expansion depends on the initial magneticfield strength inside the ME, but not significantly on the magnetic field inside the ME measured near1 au. This is an indirect evidence that CME expansion in the innermost heliosphere is driven by thehigh magnetic pressure inside the ME, while by the time the MEs reach 1 au, they are expanding due tothe decrease in the solar wind dynamic pressure with distance. We also determine the evolution of theME tangential and normal magnetic field components with distance, revealing significant deviationsas compared to the expectations from force-free field configurations as well as some evidence that thefront half of MEs expand at a faster rate than the back half.
Keywords:
Coronal mass ejections – Magnetic ejecta – Radial expansion INTRODUCTIONRadial expansion is one of the fundamental characteristics of coronal mass ejections (CMEs), described in early workusing in situ measurements (Klein & Burlaga 1982; Burlaga et al. 1982; Suess 1988). It is also clearly occurring basedon the fact that CMEs are remotely imaged as being a fraction of a solar radius wide when they erupt and are onaverage measured as being 45 solar radii (0.21 au) when they reach Earth. Associated with this increase in radialsize, the magnetic field strength inside the CME decreases as the CME propagates to larger heliocentric distances(Bothmer & Schwenn 1998). Most of what is known about the increase in radial size and decrease in magnetic fieldinside magnetic ejecta (MEs) is obtained from statistical studies of in situ measurements of different MEs at differentheliocentric distances. Thus, based on measurements by Helios, ISEE-3, IMP-8, ACE,
Wind and Voyager, usingdifferent boundaries and different subsets of CMEs, past studies have found that the radial size of an ME increases as r . to r . and the magnetic field scales as r − . to r − . (Bothmer & Schwenn 1998; Liu et al. 2005; Leitner et al.2007; Gulisano et al. 2010). This was revisited using STEREO, ACE and MESSENGER data for the solar cycle 24yielding almost the same index of radial dependency as − . ± .
19 (Winslow et al. 2015). This approach providesa measure of the average global expansion and assumes that there is a unique typical behavior of CMEs. Statistical [email protected] a r X i v : . [ phy s i c s . s p ace - ph ] J u l Lugaz et al. methods would not work well if, for example, fast CMEs always expand differently than slow ones. A different measureof the global CME expansion can be obtained in a case-by-case basis by tracking CME radial size with heliosphericimagers up to distances of about 0.5 au (Savani et al. 2009; Nieves-Chinchilla et al. 2012; Lugaz et al. 2012), whichhas revealed an expansion on the lower end of the range from statistical studies, as r . to r . . In a recent work,Al-Haddad et al. (2019) compared the index of decrease of the magnetic field with the index of increase of the MEradial size for two different simulations, finding that the initiation mechanism and the CME propagation speed do notappear to have a large influence on the ME expansion in the innermost heliosphere.Another measure of CME expansion can be obtained from the direct analysis of in situ measurements at a givenlocation, as the large majority of MEs have a decreasing speed profile. This is clearly a local measure. Figure 1shows schematic representations of the various measures of CME expansion. The expansion speed, defined as half thefront-to-back speed difference is found to vary from a few tens of km s − to as much as 250 km s − (Burlaga et al. 1982;Farrugia et al. 1993). Klein & Burlaga (1982) noted that the expansion speed is on the order of half the ambient Alfv´enspeed, meaning that expansion occurs sub-Aflv´enically. Gosling et al. (1994) and Reisenfeld et al. (2003) presented theobservations of several CMEs which were bounded by a forward-reverse shock pair. This shock pair was attributed tothe CME expansion becoming super-fast due to high pressure inside the ME. This type of over-expanding structure hasonly been reported away from the ecliptic with Ulysses observations (at latitudes greater than 22 ◦ ). In a recent study,Lugaz et al. (2017a) showed that slow CMEs may drive shocks because of their radial expansion in the ecliptic plane,although the expansion remains sub-Alfv´enic. Such shocks may form at distances of 0.2 au or greater, depending onthe rate at which the CME expansion speed and Alfv´en speed decrease (Poedts et al. 2016; Lugaz et al. 2017b).A difficulty with studying CME expansion is that the expansion speed is found to depend significantly on the CMEsize and propagation speed, with larger and faster CMEs having larger expansion speeds (Owens et al. 2005; Gulisanoet al. 2010). To solve this problem, researchers have focused on a dimensionless expansion parameter, typically theratio of the expansion to propagation speed. D´emoulin & Dasso (2009) and Gulisano et al. (2010) developed a differentformalism, in which a dimensionless expansion parameter, ζ , is defined as follows: ζ = DV c ∆ V ∆ t ∼ DS V exp V c . (1)Here, D is the heliospheric distance where the measurements are made, S = V c ∆ t is the CME size, V exp and V c are theCME expansion and center speeds, respectively, ∆ V ∆ t is the slope of the CME velocity time profile. This dimensionlessparameter scales as V − c , taking into consideration that faster and wider CMEs have higher expansion speed. Based onmeasurements in the inner heliosphere for several dozen isolated CMEs, the authors found that ζ clusters around 0.8(D´emoulin 2010). From a theoretical analysis, Gulisano et al. (2010) argued that this local measure should representthe global expansion of CMEs with the CME size growing as r ζ and the magnetic field strength decreasing as r − ζ .Note that the formula uses the slope of the velocity profile ∆ V ∆ t , which is equivalent to using the expansion speed onlyfor those cases where the velocity can be fitted linearly for the entire ME duration.The physical cause of CME expansion is still a matter of debate, although it is generally agreed that it is associatedwith pressure balance or imbalance between the ME and the solar wind. It has been proposed that CME expansionis associated with the pressure imbalance between the high pressure of the magnetically dominated ME and the lowerpressure in the solar wind (Klein & Burlaga 1982). In that sense, CME expansion is associated with over-pressure.A somewhat different explanation is that CME expansion is related to the pressure balance between the ME and thesolar wind, i.e between the ME magnetic pressure and the solar wind dynamic pressure. The fact that the solar windpressure decreases with heliospheric distance then implies that CMEs keep on expanding as they propagate outward(D´emoulin & Dasso 2009; Gulisano et al. 2010). Lastly, Suess (1988) argued that measurements of decreasing speedprofile inside MEs are associated with magnetic tension and the necessary plasma motion to maintain a force-free stateof the ME.Very few studies have investigated CME size or expansion from multiple in situ measurements in near-conjunction formore than one CME event. The exceptions are the study of Leitner et al. (2007), which focused on 7 CMEs measuredin conjunction (4 with measurements below 1 au), the recent study by Good et al. (2019), which focuses on 18 eventsand the study by Vrˇsnak et al. (2019), which focuses on 11 events during the cruise phase of MESSENGER and VEX.In particular, Good et al. (2019) found a significant difference between the power-law obtained from performing a fitof the maximum magnetic field with distance ( − . ± .
04) as compared to the average of the power-law indices ofthese 18 events ( − . ± . ME Radial Expansion From Spacecraft Conjunction Figure 1.
Schematic representation and definitions of the global and local measures of ME expansion. The idealized ME cross-section and associated magnetic field measurements are shown at two locations at different heliocentric distances. Comparingmeasurements from these two locations define the global expansion. At the second spacecraft, measurements of the plasmavelocity allow to derive various measures of the local expansion.
Here, we further dive into these datasets to investigate ME expansion. We note that near-conjunction is often takenquite loosely, as has also been done here. Angular separations for spacecraft considered in near-conjunction in thesestudies typically range from 1-20 ◦ with a few cases up to 30 ◦ with the average angular separation being ∼ ◦ in thestudy of Good et al. (2019) and ∼ ◦ for the dataset of Salman et al. (2020).To learn more about CME expansion, it is essential to compare its local measures (the ζ parameter, the expansionspeed, etc.) with global ones (how much do the CME size and magnetic field change with distance). For example, theuse of the dimensionless index of D´emoulin & Dasso (2009) is meant to take into consideration the fact that fast andwide CMEs may have a large front-to-back speed difference without having a large expansion per se . However, thisbegs the question of the cause of the large size of these CMEs. Is it related to their expansion earlier on or a large sizenear the Sun? Performing such a study has not been possible until now because it requires the investigation of CMEexpansion in both its global and local ways in a case-by-case basis for enough events to compare with past statisticalstudies. Here, we take advantage of the numerous CME events measured in conjunction between two spacecraft inthe inner heliosphere as recently presented in Salman et al. (2020) using data from MESSENGER, Venus Express(VEX), Wind and STEREO. In section 2, we quickly summarize our data and procedure. In section 4, we comparethe different measures of CME expansion with each other and with other related CME properties. In section 5, wediscuss and conclude. DATA AND METHODSSalman et al. (2020) presented 47 two-spacecraft conjunction measurements of CMEs over the first half of solarcycle 24, from 2008 to 2014 for spacecraft longitudinal separations of less than 35 ◦ , with 8 events measured atless than 5 ◦ separations and 20 at less than 15 ◦ separations. Five events were conjunction between Venus Express(VEX) and MESSENGER, 18 conjunction events occurred between MESSENGER and a spacecraft near 1 au ( Wind ,STEREO-A or STEREO-B), and 24 between VEX and a spacecraft near 1 au. Since STEREO and
Wind have plasma
Lugaz et al. instruments, we have in situ measurements of the CME speed near 1 au for these 42 CMEs in addition to magneticfield measurements at two different distances. For the five conjunctions events between MESSENGER and VEX, wedo not have any plasma measurements. Our analysis thereafter focuses on the 42 events with plasma measurementsnear 1 au. Because Mercury’s heliocentric distance (and therefore MESSENGER’s) varies between 0.31 and 0.47 au,whereas Venus stays at 0.72-0.73 au, we have measurements over distances varying from a factor of 1.3 (Venus toSTEREO-A) to a factor of 3.2 (Mercury at perihelion to STEREO-B).The magnetic field decrease with heliospheric distance for this dataset is presented by Salman et al. (2020) who founda decrease of the maximum field, B max , inside the ME with an index of − . ± .
25. Although most events have gapsin measurements corresponding to the time when MESSENGER or VEX are inside their planetary magnetosphere, wecan do the same study with the average magnetic field, B av for which we find an index of − . ± .
32 excluding the5 MESSENGER-VEX conjunctions. For each conjunction event, we also calculate the quantity α B (see for exampleDumbovi´c et al. 2018): α B = log ( B /B )log ( r /r ) , where indices 1 and 2 correspond to the first (closer to the Sun) and second (further away from the Sun) spacecraft,respectively. We do so for both the maximum magnetic field ( α Bmax ), the average magnetic field ( α Bav ), as well asthe maximum value of the tangential ( T ) and normal ( N ) magnetic field components inside the ME ( α BT and α BN ).In addition, near 1 au, we derive local measures of the ME radial expansion: 1) the dimensionless expansion parameter ζ fit using the procedure of Gulisano et al. (2010), i.e. by performing a linear fit on the velocity data to derive ∆ V / ∆ t .We also calculate 2) the expansion speed V exp , 3) the ratio of the expansion speed to the CME speed V exp /V center , and4) ζ mes using the measured value of ∆ V = 2 V exp in equation (1) rather than the fit to the velocity data. We also useor derive associated CME properties: its initial speed from coronagraph (as listed in Salman et al. 2020), its size near1 AU (using the average CME speed), and the maximum and average magnetic field inside the ME. RESULTS: SPECIFIC EVENTSMost of the best conjunction events have been studied in detail in previous work. Here, we present one additionalevent to illustrate our technique and summarize results for three previously published events. The four events wehighlight are among the eight best conjunctions (separations of less than 5 ◦ ) with data near 1 au. The results of theanalysis described below for these four events are listed in Table 1.3.1. The May 1–4, 2013 CME event (event 14-2013 in Salman et al. 2020) is a conjunction between MESSENGER (at0.36 au) and STEREO-A (at 0.96 au) when the longitudinal separation between the two spacecraft was only ∼ . ◦ .The measurements at MESSENGER and STEREO-A are shown in Figure 2. This was a moderately fast event witha coronagraphic speed of 700 km s − and a maximum ME speed near 1 au of 570 km s − . In most cases, solar windplasma measurements are not available at Mercury with MESSENGER. The maximum ME magnetic field of 132 nTat MESSENGER and 22 nT at STEREO-A results in a value of α Bmax = − .
84, which is relatively typical. Theexponent for the average magnetic field is very similar at α Bav = − .
89. The speed profile at 1 au is complex and weconsider that a linear trend in the velocity can only be found for the front 30% of the ME. Using this limited period, alinear fit to the velocity profile implies that ζ fit = 1.7. The expansion speed as measured from maximum to minimumis 62 km s − and the ME center speed is about 485 km s − . Using the measured V exp , we can derive ζ mes = 0 .
61. Theratio of expansion to center speeds is ∼ B max = 102 nT and α B = − .
57. The presence of fast-forward shocks at the back of MEs wasdiscussed in Lugaz et al. (2015). Such a shock/discontinuity was not observed at STEREO-A. Based on past work,this raises two possibilities: i) the shock fully propagated through the ME before the ME impacting STEREO-A. Inthat case, the period of compression by the shock is expected to be followed by a period of over-expansion (Gulisanoet al. 2010; Lugaz et al. 2012). Depending on the timing of this exit, the ME global and local measures of expansionmay be affected. ii) The shock dissipated as it propagated inside the ME (Farrugia & Berdichevsky 2004; Lugaz et al.2007) and only the back half got affected. In both cases, the back half of the ME may have been compressed, resulting
ME Radial Expansion From Spacecraft Conjunction Figure 2. left ) and MESSENGER ( right ). A linear trend in the velocity can befound in about the first 30% of the ME at STEREO-A. This is used to derive ζ fit . The maximum and average of the magneticfield magnitude as well as the maximum and minimum of B T and B N are used to derive various exponent decrease α . The redline marks the shock arrival at STEREO-A and MESSENGER, the blue lines mark the ME boundaries with dashed lines usedwhen the boundary’s location is not certain. For the MESSENGER data, the dashed red line shows a shock propagating insidethe ME. in the flat velocity profile in the back measured near 1 au. At MESSENGER, there is no clear driver for this shockas the magnetic field strength goes back to normal values a few hours after the shock. At such, it is unlikely that theME measured near 1 au is the result of the merging of two CMEs.The decrease of the tangential and normal magnetic field components for the front of the ME (positive values) is α Tfront = − .
40 and α Nfront = − .
68. The back (negative values) for which the peak occurs after the shock at Mercuryare α Tback = − .
41 and α Nback = − .
29. This shows that the normal (north-south) component of the magnetic fielddecreased a bit faster than the tangential (east-west) component, but also highlights how this detailed analysis maybe affected by the presence of shocks and “datagaps” in MESSENGER measurements associated with magnetosphericcrossings. 3.2.
Other Events
Good et al. (2015) and Salman et al. (2020) presented a different conjunction that occurred on November 4–8, 2011(event 8-2011) with an initial speed of 750 km s − and a maximum ME speed near 1 AU of 440 km s − . Although adifferent section of the same event also impacted Venus, the best conjunction is between MESSENGER and STEREO-B ( ∼ . ◦ longitudinal separation). For this event, α Bmax = − .
93 and α Bav = − .
80, but there is a large expansionspeed at 1 AU of ∼
85 km − corresponding to ζ fit = 1 . ζ mes = 0 .
95 and a ratio of the expansion to the center MEspeed of 18%. Once again, local and global measures of expansion disagree.
Lugaz et al.
Table 1.
Examples from past studies and Figure 2. Results with a ∗ indicate cases for which the peak is likely to have occurredduring a magnetospheric path of MESSENGER and is therefore likely missed. Values in parentheses for B T or B N correspondto the decrease for that component of the magnetic field in the back half of the ME.Event Sep. V init α Bmax α Bav α BT α BN ζ fit ζ mes V exp /V c ◦
950 km s − − . − . − . − . − . − .
5) 1.5 0.95 0.189-2011 4.6 ◦
760 km s − − . − . − . − . − . − .
2) 0.19 0.15 0.0114-2013 2.9 ◦
700 km s − − . − . − . − . − . − .
3) 1.7 0.61 0.1321-2013 3.1 ◦
700 km s − − . − . − . ∗ − . − .
1) 0.67 0.51 0.10
Winslow et al. (2016) presented a complex conjunction event (event 9-2011) between MESSENGER (at 0.42 au)and STEREO-A ( ∼ . ◦ longitudinal separation) on December 30, 2011 – January 1, 2012 with an initial speed of950 km s − and a maximum ME speed near 1 au of 630 km s − . For this event, α Bmax = − .
99 and α Bav = − . − , ζ fit = 0 .
19 and ζ mes = 0 .
15. In this case, the decrease of the magnetic field inside the ME with distance is typical,but the bulk speed profile at 1 AU indicates a lack of expansion near 1 AU. This case is somewhat unusual because ofthe complex interaction with the heliospheric current sheet that is found to be engulfed inside the ME at 1 AU. Theratio of expansion to center speeds is of the order of 1%.Another event (event 21-2013) was recently discussed in Lugaz et al. (2020) for a conjunction between MESSENGERand L1 ( ∼ . ◦ longitudinal separation) on July 11–14, 2013 with an initial speed of 600 km s − and a maximum MEspeed near 1 au of 500 km s − . This long-duration event is found to have α Bmax = − .
58 and α Bav = − .
38 and
Table 2.
Average values and 1- σ standard deviations obtained in this study and comparison to past studies. The first fourquantities are obtained in our study by measuring the magnetic field at two spacecraft in conjunction, while for the past studies,these are typically from fits to different MEs measured at different heliocentric distances. The seven other quantities are obtainedfrom measurements near 1 au. F05: Farrugia et al. (2005), L07: Leitner et al. (2007), W15: Winslow et al. (2015), G19: Goodet al. (2019), W05: Wang et al. (2005), L05: Liu et al. (2005), G10: Gulisano et al. (2010), D08: D´emoulin et al. (2008), RC10:Richardson & Cane (2010), J18: Jian et al. (2018), NC18: Nieves-Chinchilla et al. (2018) and L18: Lepping et al. (2018). Thedata source in the inner heliosphere is indicated in parentheses (H: Helios, P: Pioneer Venus Orbiter, M: MESSENGER, V: Venus Express ). For α Bmax , we list two values for G19, the first one using the same procedure as done here but for 13 eventsand the second one using a fitting procedure.Quantity Average ± σ Past Results Source α Bmax − . ± ± ± σ ), F05(H), L07(H,P), W15(M)-1.34 ± ± σ ) G19(M,V) α Bav − . ± ± ± ± σ ) G10(H), W15(M) α BT − . ± α BN − . ± ζ fit ± ± ± ζ mes ± V exp (km s − ) 32 ±
42 31 ±
3, 62 ±
3, 28 RC10, J18, NC18 V center (km s − ) 449 ±
131 476 ±
6, 445, 434, 436 RC10, J18, NC18, L18 V exp /V center ± S ME (AU) 0.29 ± ± ± < B ME > (nT) 10.8 ± ± ζ fit = 0 .
67 for an expansion speed of about 50 km s − corresponding to ζ mes = 0 .
51 and V exp /V center ∼ ◦ , the local and global measuresof expansion do not necessarily agree. In the following section, we look at overall results for all CMEs and comparethe various measures of expansion with each others. ME Radial Expansion From Spacecraft Conjunction RESULTS: STATISTICS4.1.
Average Values
Table 2 shows the statistics of the values of the different measures of CME expansion and other CME properties, aswell as comparison to previous studies in the inner heliosphere (excluding other studies with Ulysses or Voyager datapast 2 au). Throughout, we give the 1- σ standard-deviation as an error bar when quoting a value.For α B max , the average is − . ± .
84 for the full dataset but − . ± .
43 for the conjunctions between MESSENGERand 1 au and − . ± .
06 for the conjunctions between VEX and 1 au. In all cases, this is a similar average as comparedto the results from statistical studies but with a much larger standard-deviation, as for example Winslow et al. (2015)found a 3- σ value of ± .
19. Combining all past studies, a range for the exponent decrease of B max can be obtainedas − . ± .
4. Only 20 out of the 42 events studied here are within this range. For α Bav , we find an average of − . ± .
90. The average value for ζ is 0.86 ± .
83, comparable to past studies. We note that we are able to identifya linear trend in the velocity profile for 82% ±
22% of the ME duration (for 31 events, a trend is identified for morethan 60% of the ME duration). In fact, the event highlighted in Figure 2 is the one for which the linear trend is theleast clear. The average expansion speed is 32 ±
44 km s − (or 39 ±
35 km s − if excluding three contracting events)and that of the dimensionless expansion is 0.067 ± .
09 (0.083 ± .
06 if excluding these three events). The average sizeof the MEs near 1 au is 0.29 ± .
14 au, comparable to that found from all
ACE
MEs by Richardson & Cane (2010).That number is larger than the canonical 0.21 au from Lepping et al. (2018) but the latter is obtained for a force-freefit to the data, whereas our number and that by Richardson & Cane (2010) are simply derived by integrating the solarwind speed with time during the ME passage. For all quantities, the average values are within the typical ranges frompast studies, highlighting that our dataset is not biased.4.2.
Comparison of Local and Global Measures of CME Expansion
In the top panels of Figure 3, we show plots of α Bmax and α Bav as compared to ζ fit and ζ mes as well as the valueswhen the first spacecraft is MESSENGER rather than VEX since the α values have less variability when the former isthe first spacecraft rather than the latter. The symbols are color-coded with the spacecraft angular separation and thetop panels show the line α = 2 ζ , which is the expected trend. The data is un-correlated for ζ fit (obtained by fitting tothe slope of the velocity), while there is a very weak correlation with ζ mes (calculated using the measured expansionspeed) with the highest correlation coefficient, r = 0 . α Bmax and α Bav are compared to V exp and V exp /V center . This again comparesglobal quantities of CME expansion (in the y -axis) to local quantities near 1 au of the CME expansion (in the x -axis).The largest correlation coefficient is found between the α index for the average magnetic field and the expansion speedand is r = 0 . α are associated with typical values of ζ around its average of 0.7.4.3. Correlation of Global Measures of CME Expansion with Other CME Properties
We extend the analysis of local and global measures of expansion to determine whether other CME propertiesare correlated with CME global expansion. We focus on the CME initial speed, obtained from the best-observingcoronagraph, as explained in Salman et al. (2020), the CME final speed measured near 1 AU, as well as the CMEmagnetic field strength as measured at various distances. Figures 4 and 5 show the results for α as compared to theME velocity and magnetic field, respectively, in the same format as Figure 3.The ME expansion is only weakly correlated with the CME initial speed, with faster CMEs expanding more rapidlyin the inner heliosphere. This correlation remains present near 1 au for α Bav as compared to the CME front andcenter speeds. It is only a weak correlation but reflects that faster CMEs do expand more strongly in a statisticalsense. We note that the dimensionless analysis of Dasso et al. (2009) and Gulisano et al. (2010) results in ζ beingapproximately independent of the CME speed, but here we find a weak correlation between α av and the CME speed.It is possible that a stronger correlation would exist if the speed was measured at the first spacecraft or if the expansionwas calculated for distances closer to the Sun, where expansion may be more related to the initial characteristics ofthe CME.We then direct our attention to the correlation between α and the magnetic field inside the ME at various distances.When we compare the α parameter with magnetic field measurements, we only compute the correlation of α Bav with
Lugaz et al.
Figure 3.
Global ( y -axis) vs. local ( x -axis) measures of CME expansion. The top panels show the index decrease of themagnetic field, α , as compared to the dimensionless expansion parameter near 1 AU, ζ . The line show the expected value of α = − ζ . The bottom panels show α as compared to the ME expansion speed near 1 AU (left) and the ratio of ME expansionto center speeds (right). The thin line shows the linear relation for the best fit: − α Bav = 1 .
66 + 0 . V exp . All data points arecolor-coded with the angular separation between the two spacecraft with the scale in ◦ given on the right-hand colorbar. B max and of α Bmax with B av . This way, the values of the magnetic field used to calculate α are not compared with thesame values measured at various locations. We note however, that B max and B av are obviously very well correlated(correlation coefficient ∼ α with the magnetic field (averageor maximum) measured near 1 au with a correlation coefficient below 0.15, whether or not it is corrected for thedifference in heliocentric distance between the various spacecraft (see below for details). However, we find a muchstronger correlation with the magnetic field measured by Venus Express, with a correlation coefficient of 0.62-0.65(bottom left panel).Correlating the magnetic field measured by MESSENGER with the CME expansion is not straight-forward, becauseMESSENGER heliocentric distance in our sample varies between 0.308 and 0.466 au. For a typical decrease of themagnetic field as r − . , this means that the magnetic field would decrease by more than a factor of 2 between thesetwo distances. In comparison, VEX is always between 0.72 and 0.73 au and the variation in magnetic field strengthbetween STEREO-A at 0.96 au and STEREO-B at 1.09 au is only by a factor of 1.25. To correct for the variationin the heliocentric distance of MESSENGER, we scale all measurements to 0.308 au (the measurement made at thelowest heliocentric distance) using the α value obtained for this particular CME. The results show a strong correlation(bottom right panel of Figure 5). It should be noted that we use (for example) the value of α Bav obtained for a
ME Radial Expansion From Spacecraft Conjunction Figure 4.
Global measures of CME expansion ( y -axis) vs. CME speed. The left panel shows the index decrease of the magneticfield, α , as compared to the initial plane-of-sky coronagraphic speed. The thin line shows the linear relation for the best fit (in alog-linear plot): − α Bav = 1 .
40 + 0 . V init . The right panel shows α Bav as compared to the final front and center ME speedsnear 1 AU. The thin line shows the linear relation for the best fit: − α Bav = 0 .
81 + 0 . V front . All data points are color-codedwith the angular separation between the two spacecraft with the scale in ◦ given on the right-hand colorbar. particular CME to scale the value of B av measured for this CME by MESSENGER to 0.308 au and compare it with α Bmax . As such, we use fully separated measurements to determine the correlation. Lastly, we scale all VEX andMESSENGER measurements to 0.308 au and obtain very significant correlations between the scaled value of B in theinner heliosphere and α , the expansion index (top right panel of Figure 5).We interpret these results as follows: in the innermost heliosphere, there is clear positive correlation between theME maximum magnetic field and the expansion index, i.e. , that MEs with higher internal magnetic pressure in theinnermost heliosphere expand more on their way to 1 au. However, near 1 au, there is no relation between the internalmagnetic pressure and how much expansion occurred.In addition, the range of ME average magnetic fields is narrower near 1 au than near 0.72 au (at VEX) and atMESSENGER. In our sample, the average ME magnetic field at 1 au is 10.3 nT ±
33% (with STEREO measurementsscaled to 1 au), at VEX, it is 18.6 nT ±
36% and at MESSENGER it is 87 nT ±
42% scaled to 0.308 au (62 nT ± Evolution with Distance of Magnetic Field Components Inside MEs
Vrˇsnak et al. (2019) investigated how the fitted magnetic field inside MEs and the radial size of MEs vary withdistance and discussed the implications of their study for the self-similar expansion of MEs. They concluded that,for individual cases, reconnection between the ME and the solar wind and/or pancaking of the ME cross-sectionis necessary to understand the evolution of the ME size as compared to the evolution of the magnetic field. In aprevious work, Leitner et al. (2007) noted that the expected difference in the decrease rate with distance of the axialand azimuthal components of the magnetic field may create differences in the trend found for the inner and outer0
Lugaz et al.
Figure 5.
Global measures of CME expansion ( y -axis) vs. ME magnetic fields measured or scaled to various distances. Thetop left panel shows the index decrease of the magnetic field, α , as compared to the ME magnetic field strength measured near1 AU. The top right panel shows α as compared to the ME magnetic field strength measured by the spacecraft closest to theSun (VEX or MESSENGER) and scaled to 0.308 au (see text for details). The bottom panels show α as compared to the MEmagnetic field measured by VEX (left) and measured by MESSENGER and scaled to 0.308 au (right). The colorbars are thesame as in Figures 3 and 4. heliosphere. Good et al. (2019) discussed the change in orientation of the 18 CMEs measured in conjunction that theystudied, finding a tendency towards lower-inclined MEs at the outer spacecraft compared to the first spacecraft. Thisimplies a (small) difference in the way different magnetic field components change with distance.We note that for a force-free field with self-similar expansion, the axial magnetic field is expected to vary withdistance as r − , whereas the poloidal field should vary as r − . In Lugaz et al. (2020) for the 2013 July 10-13 CME, wefound that a uniform decrease of the magnetic field components as r − . was a better fit to the data than a separatefit for the y (axial) or z (poloidal) components of the magnetic field. Here, we continue this analysis for the 42 CMEsmeasured in conjunction between two spacecraft.Because MESSENGER and VEX were planetary missions, there are significant “data gaps” in the IMF measurementscorresponding to the time when the spacecraft were in the planetary magnetosphere. In addition, the magnetic fieldsinside MEs have been reported to significantly rotate in the inner heliosphere in some cases (e.g., see Nieves-Chinchillaet al. 2012; Winslow et al. 2016) and it is unclear how this should be considered when comparing the axial or poloidalfields measured by these spacecraft with those measured near 1 au. As such, we compare the tangential and normalcomponents of the magnetic field measured at the two spacecraft in the RT N coordinate system. We focus on theextrema of the variation of the magnetic field components.
ME Radial Expansion From Spacecraft Conjunction Figure 6.
Left: Average of the expansion indices of the positive and negative B N component vs. average of the expansionindices of the positive and negative B T components. The lines shows the 1-to-1, 1-to-2 and 2-to-1 values as well as the expected − − B T and B N in the front and back of the MEs. For an ME that has a clear low (resp. high) inclination, the B T (resp. B N ) component typically keeps the same signthroughout the ME interval. In addition, the expansion of the front and back half of the ejecta may occur at differentrates. For example, in event 21-2013, discussed in Lugaz et al. (2020), B T is always positive, while B N varies frompositive to negative (NWS ME following the classification of Bothmer & Schwenn 1998). For this event, as shownin Table 1, the B N positive component (at the front) decreases with an index of − .
4, whereas the B N negative (atthe back) decreases with an index of − .
1. We therefore calculate the average of the indices for the positive andnegative extrema of one component, and compare these. For the 21-2013 event, this means comparing the index of − . B T unipolar component with − .
25 for the average of the B N indices. This shows that, although this isa low-inclined cloud, the axial and poloidal fields do not expand with a 1-to-2 ratio, but have approximately the samerate of expansion. We perform the same analysis for all MEs and these averages for the index decrease of the B T and B N components inside the ME are plotted in the left panel of Figure 6.This Figure shows that there is no ME for which one component decreases as r − while the other decreases as r − ,which would be expected for the force-free expansion of a low or high inclined ME. There are a few cases for whichthis is approximately true. In fact for most MEs, the expansion index of the normal and tangential components agreewith each other. The average of the expansion index of B T and B N are nearly identical (see Table 2 and the ratioof α BT to α BN is 1.09 ± ◦ ,which would imply that the normal and tangential components decrease similarly in a force-free model. This is highlyunlikely; if nothing else, the four events described in Section 3 include MEs with a low inclination. In addition, sucha situation should result in indices of both components around − .
5, whereas we find a cluster of MEs for which bothcomponents decrease approximately as r − .Lastly, we compare the expansion of the components in the front half of the MEs with that in the back half. Theresults are plotted in the right panel of Figure 6. It shows a bias towards the expansion in the front of the ejecta tobe stronger than the expansion at the back. Note that we have reliable exponents only for 34 pairs (front and back)of magnetic field components, and that these are dominated by conjunctions involving MESSENGER data (28 casesvs. six for VEX data). The ratio of the front-to-back expansion is 1.57 ± B T and B N ,the exact position of the boundaries is not expected to influence the results. This is somewhat consistent with thefindings of Janvier et al. (2019) that showed that the profile of the magnetic field inside MEs is more peaked towardsthe front at MESSENGER and more symmetric at 1 au. This would result in the front half to show more expansionthan the back half of MEs as found here. This result may be associated with the presence of a sheath region in frontof the ME that allows the front part of the ME to expand relatively freely. On the contrary, the expansion of the2 Lugaz et al.
Figure 7.
Schematic representation of the expansion of two MEs. In the inner heliosphere, one ME (orange) expands moreslowly than the other (blue) until both reach total pressure balance with the solar wind (bottom row). Afterwards, they expandwith the same rate, dictated by the solar wind expansion. Both MEs have α = − . α = − . α = − .
2. The combined α from 0.3 to 1 auare − . − . In situ measurements (top row) near 1 au do notreflect what happened in the innermost heliosphere, while measurements below ∼ . α of various MEs (middle row) tend towards the solar wind value (green curve) as the MEs approach1 au with the orange and blue curves representing the MEs shown below and other colors for other potential behaviors. back part of the ME may be hindered by the presence of the ME wake with speed comparable (or sometimes slightlyhigher) than the back of the ME. The presence of fast solar wind streams behind MEs may also result in MEs beingsomewhat compressed in the back, and would thus still be consistent with these results. The presence of fast streamsbehind MEs near 1 au is a relatively frequent occurrence. We note that this cannot be explained by aging as the backpart of the ME is older than the front when it passes over a spacecraft and it is the section of the ME which has hadmost time to expand. This finding, if confirmed, further complicates the notion of force-free and self-similar expansionof MEs as there might not be a balance of the magnetic field at all time throughout the ME propagation. DISCUSSION AND CONCLUSIONSIn this work, we have used in situ measurements of 42 CMEs made in conjunction by two spacecraft among MESSEN-GER, Venus Express,
Wind , STEREO-A and STEREO-B during solar cycle 24 to compare global and local measuresof ME expansion. In terms of global measures, we have focused on the index of the decrease of the magnetic field withdistance, α . In terms of local measures, we have examined the expansion speed and various dimensionless parameters,primarily ζ from D´emoulin & Dasso (2009) as calculated near 1 au. We have also compared the global expansionwith local properties of CMEs, its initial and final speed and magnetic field strength. Our sample, in terms of averageproperties of the CMEs, appears typical when compared to the average properties from larger samples measured near1 au (Richardson & Cane 2010; Jian et al. 2018).We have found that the global and local measures of CME expansion are, at best, only weakly correlated, indicatingthat measurements near 1 au do not reflect the expansion of CMEs between ∼ ME Radial Expansion From Spacecraft Conjunction ∼ ∼ ζ ) do not reflect processes that occurred below ∼ α , is dominated by what happens in the innerheliosphere. In the example in Figure 7, both MEs have α = − . α = − . α = − . α from 0.3 to1 au are − . − .
3, even though the ME magnetic field at 1 au is the same for both cases with a value of 13.5 nT.While this scenario fits with the various findings in this work, to be fully tested, it would require i) more conjunctionevents involving three or more spacecraft, and ii) plasma measurements, especially of the velocity, in the inner helio-sphere (below 0.95 au) to test the prediction that the ζ parameter may be better correlated with α in the innermostheliosphere.In addition, we have found some evidence from the evolution of the tangential and normal components of the magneticfield inside MEs between the two spacecraft that MEs do not maintain force-free conditions while they expand. Thisconclusion has been obtained without performing fitting of the magnetic field measurements, which would require tomake assumptions regarding the morphology of the magnetic field inside MEs. In addition, fitting methods have beenfound to often disagree regarding ME orientation (Al-Haddad et al. 2013).Lastly, we have found evidence that the front of the ME expands faster than the back. This might be consistentwith the back half of the ME being overtaken by the solar wind behind it. Such a scenario would hinder the MEexpansion in its back half. This finding is consistent with the fact that many in situ measurements within MEs, suchas those presented in Figure 2, have a decreasing speed profile in the front part of the ME and a constant ME speedequal to the solar wind speed in the back of the ME. This indicates that the ME expansion in the ecliptic plane is notable to continue beyond the point where the ME back speed equals the solar wind speed. This is also consistent withthe lack of reverse shocks measured at the back of MEs in the ecliptic plane, contrary to what occurs at the back ofstream interaction regions or MEs at high latitudes (Gosling et al. 1998).Some of these results could be further tested if we had multi-spacecraft measurements of CMEs made at approxi-mately the same heliocentric distance. This would allow us to compare different local measures of the CME expansion(expansion speed, ζ , etc.) to determine how they vary through different crossings within the same ME. Such multi-spacecraft measurements will be possible when STEREO-A comes back to the proximity of the Sun-Earth line in2023-2024, but this will only provide about 11 months of potential measurements within 10 ◦ from the Sun-Earth line.Lugaz et al. (2018) highlighted differences between spacecraft measurements for angular separation of ∼ . ◦ ; howeverthe maximum magnetic field strength remained very consistent between two spacecraft even when the componentsmeasured by the two spacecraft showed significant differences. If such differences between MEs are common, the resultsabout the expansion of various magnetic field components may be affected. This highlights the need for a dedicatedmission providing multi-point measurements of MEs in the inner heliosphere.ACKNOWLEDGMENTSThis work has been made possible by the following grants: NASA NNX15AB87G, 80NSSC20K0700, 80NSSC17K0556and 80NSSC20K0431 and NSF AGS1435785. RMW acknowledges support from NASA grant 80NSSC19K0914 andNSF grant AGS1622352. CJF acknowledges support from Wind grant 80NSSC19K1293. All the data analyzed in thisstudy are publicly available. MESSENGER and VEnus Express data are available on the Planetary Data System (https://pds.jpl.nasa.gov) while other data are available from the CDAWeb (https://cdaweb.sci.gsfc.nasa.gov/index.html/)REFERENCES
Al-Haddad, N., Lugaz, N., Poedts, S., et al. 2019,Astrophys. J., 884, 179, doi: 10.3847/1538-4357/ab4126 Al-Haddad, N., Nieves-Chinchilla, T., M¨ostl, C., et al. 2013,Solar Phys., 284, 129, doi: 10.1007/s11207-013-0244-5 Lugaz et al.
Bothmer, V., & Schwenn, R. 1998, Annales Geophysicae,16, 1, doi: 10.1007/s00585-997-0001-xBurlaga, L. F., Klein, L., Sheeley, N. R., et al. 1982,Geophys. Res. Lett., 9, 1317Dasso, S., Mandrini, C. H., Schmieder, B., et al. 2009, J.Geophys. Res., 114, 2109, doi: 10.1029/2008JA013102D´emoulin, P. 2010, Twelfth International Solar WindConference, 1216, 329, doi: 10.1063/1.3395866D´emoulin, P., & Dasso, S. 2009, Astron. Astrophys., 507,969, doi: 10.1051/0004-6361/200912645D´emoulin, P., Nakwacki, M. S., Dasso, S., & Mandrini,C. H. 2008, Solar Phys., 250, 347,doi: 10.1007/s11207-008-9221-9Dumbovi´c, M., Heber, B., Vrˇsnak, B., Temmer, M., &Kirin, A. 2018, Astrophys. J., 860, 71,doi: 10.3847/1538-4357/aac2deFarrugia, C., & Berdichevsky, D. 2004, AnnalesGeophysicae, 22, 3679Farrugia, C. J., Burlaga, L. F., Osherovich, V. A., et al.1993, J. Geophys. Res., 98, 7621, doi: 10.1029/92JA02349Farrugia, C. J., Leiter, M., Biernat, H. K., et al. 2005, inESA Special Publication, Vol. 592, Solar Wind 11/SOHO16, Connecting Sun and Heliosphere, ed. B. Fleck, T. H.Zurbuchen, & H. Lacoste, 723Good, S. W., Forsyth, R. J., Raines, J. M., et al. 2015,Astrophys. J., 807, 177,doi: 10.1088/0004-637X/807/2/177Good, S. W., Kilpua, E. K. J., LaMoury, A. T., et al. 2019,J. Geophys. Res., 124, 4960, doi: 10.1029/2019JA026475Gosling, J. T., Bame, S. J., McComas, D. J., et al. 1994,Geophys. Res. Lett., 21, 237Gosling, J. T., Riley, P., McComas, D. J., & Pizzo, V. J.1998, J. Geophys. Res., 103, 1941,doi: 10.1029/97JA01304Gulisano, A. M., D´emoulin, P., Dasso, S., Ruiz, M. E., &Marsch, E. 2010, Astron. Astrophys., 509, A39,doi: 10.1051/0004-6361/200912375Janvier, M., Winslow, R. M., Good, S., et al. 2019, Journalof Geophysical Research (Space Physics), 124, 812,doi: 10.1029/2018JA025949Jian, L. K., Russell, C. T., Luhmann, J. G., & Galvin,A. B. 2018, Astrophys. J., 855, 114,doi: 10.3847/1538-4357/aab189Klein, L. W., & Burlaga, L. F. 1982, J. Geophys. Res., 87,613, doi: 10.1029/JA087iA02p00613Leitner, M., Farrugia, C. J., M¨oStl, C., et al. 2007, Journalof Geophysical Research (Space Physics), 112, A06113,doi: 10.1029/2006JA011940 Lepping, R. P., Wu, C.-C., Berdichevsky, D. B., & Szabo,A. 2018, Solar Phys., 293, 65,doi: 10.1007/s11207-018-1273-xLiu, Y., Richardson, J. D., & Belcher, J. W. 2005, Planet.Space Sci., 53, 3, doi: 10.1016/j.pss.2004.09.023Lugaz, N., Farrugia, C. J., Davies, J. A., et al. 2012,Astrophys. J., 759, 68, doi: 10.1088/0004-637X/759/1/68Lugaz, N., Farrugia, C. J., Smith, C. W., & Paulson, K.2015, J. Geophys. Res., 120, 2409,doi: 10.1002/2014JA020848Lugaz, N., Farrugia, C. J., Winslow, R. M., et al. 2018,Astrophys. Journ. Lett., 864, L7,doi: 10.3847/2041-8213/aad9f4—. 2017a, Astrophys. J., 848, 75,doi: 10.3847/1538-4357/aa8ef9Lugaz, N., Manchester, W. B., Roussev, I. I., T´oth, G., &Gombosi, T. I. 2007, Astrophys. J., 659, 788,doi: 10.1086/512005Lugaz, N., Temmer, M., Wang, Y., & Farrugia, C. J. 2017b,Solar Phys., 292, 64, doi: 10.1007/s11207-017-1091-6Lugaz, N., Winslow, R. M., & Farrugia, C. J. 2020, J.Geophys. Res., 125, e2019JA027213,doi: 10.1029/2019JA027213Nieves-Chinchilla, T., Colaninno, R., Vourlidas, A., et al.2012, J. Geophys. Res., 117, 6106,doi: 10.1029/2011JA017243Nieves-Chinchilla, T., Vourlidas, A., Raymond, J. C., et al.2018, Solar Phys., 293, 25,doi: 10.1007/s11207-018-1247-zOwens, M. J., Cargill, P. J., Pagel, C., Siscoe, G. L., &Crooker, N. U. 2005, J. Geophys. Res., 110, 1105,doi: 10.1029/2004JA010814Poedts, S., Pomoell, J., & Zuccarello, F. P. 2016, inAmerican Institute of Physics Conference Series, Vol.1714, American Institute of Physics Conference Series,030002, doi: 10.1063/1.4942571Reisenfeld, D. B., Gosling, J. T., Forsyth, R. J., Riley, P.,& St. Cyr, O. C. 2003, Geophys. Res. Lett., 30, 8031,doi: 10.1029/2003GL017155Richardson, I. G., & Cane, H. V. 2010, Solar Phys., 264,189, doi: 10.1007/s11207-010-9568-6Salman, T. M., Winslow, R. M., & Lugaz, N. 2020, J.Geophys. Res., 125, e2019JA027084,doi: 10.1029/2019JA027084Savani, N. P., Rouillard, A. P., Davies, J. A., et al. 2009,Annales Geophysicae, 27, 4349Suess, S. T. 1988, J. Geophys. Res., 93, 5437,doi: 10.1029/JA093iA06p05437Vrˇsnak, B., Amerstorfer, T., Dumbovi´c, M., et al. 2019,Astrophys. J., 877, 77, doi: 10.3847/1538-4357/ab190a
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