Database Completeness

One of the goals for this database is to ensure that there are no restrictions on other variables such as MLT or altitude. We want this database to be a set of orbits that show a complete picture of the data, so we need to verify that there have been no unintended constraints placed on other variables.

Ideally, the final database of FAST orbits would provide uniform sampling over all

Figure 2.4: Final set of 10,679 FAST orbits (dark gray) shown as a fraction of the 25,961 available orbits after the first reduction (light gray), plotted across night side MLTs.

night side MLTs. The three criteria for determining a useful orbit have no dependence on night side MLT other than requiring the boundaries to occur at similar MLTs, thus requiring the orbit trajectory to be along a magnetic meridian. Figure 2.5 shows the distribution of data collected in all 10,679 fast passes through night side MLTs.

There are three features to note: a high concentration of data collected at 90^◦ latitude, a high concentration of data collected near 80^◦ latitude, and a lower con-centration of data collected near dawn and dusk. The high concon-centration of data near 90^◦ is a result of the selection criteria (as all orbits have north-south trajectories and will all pass close to 90^◦ latitude) as well as the convergence of MLT sections near the pole (the same number of orbits are pushed through smaller MLT wedges closer to the pole, resulting in high concentrations). The lower concentration of data collected near dusk and dawn is mainly due to the inability to identify boundaries at those MLTs, most likely due to bleeding effects from particles on the day side.

This feature can also be seen in Figure 2.4 as the decrease in orbits near the edges of

Figure 2.5: Distribution of each data point collected by each of the 10,679 FAST orbits, shown across night side MLTs. The selected FAST overflights provide a suf-ficiently uniform sampling of all night side MLTs. The concentric circles represent magnetic latitude with the outer ring starting at 55^◦ and stepping by 5^◦.

satellite reaching its peak altitude. The collection rate is constant throughout the orbit, so as the spacecraft reaches apogee its motion across latitudes slows and more data is collected in that region.

The sampling across altitudes is much different than that across night side MLT.

The majority of data will be collected at altitudes approaching 4200km, as the FAST apogee occurs near the magnetic poles. Figure 2.6 shows the distribution of data collected as a function of altitude and magnetic latitude. We can also identify in Figure 2.6 a high concentration region near 80^◦ latitude at various altitudes. This corresponds to the high concentration region identified at those same latitudes in Figure 2.5.

Overall, the database supplies a sufficient sampling of data across all night side MLTs and altitudes. Any nonuniformity in the data is a result of the orbit trajetory and is not affected by the selection criteria. The end result is a database of 10,679 FAST orbits containing a complete record of ESA data, including the locations of all

Figure 2.6: Distribution of data points collected by each of the 10,679 FAST orbits, as a function of altitude and magnetic latitude. The high concentration of data near high altitudes is due to the satellite apogee, resulting in a large amount of data collected in a small region of latitude.

three boundaries.

2.3.3 Magnetospheric Activity Data

The final piece of data to be collected for this study was the information regard-ing the magnetospheric activity durregard-ing each of the overflights in the database. The activity is described using the Kp index, ranging from 0-9+, where 0 is little to no activity and values over 5 indicate intense storm conditions. Values for the Kp in-dex during each FAST overflight were collected using the data set from the OMNI 2 spacecraft. Included in this data set are measurements of hourly averages of various solar parameters, including magnetic field values, bulk plasma information, sunspot numbers, and, of course, Kp values. The Kp index for each overflight was determined by taking the hourly averaged value from OMNI 2 at the time that FAST recorded the IB position. Since FAST passes through all three boundaries (IB, Io, and PCB)

Chapter 3

Analysis and Results

The full data set collected using the criteria outlined in Chapter 2 is plotted in Figure 3.1. The plot provides an overall picture of the MLT coverage and the general trend in latitudinal positions of each recorded boundary. Another representation of the full data is shown in Figure 3.2, where each boundary has been binned and plotted as a function of its magnetic latitude in degrees. These two plots together are good representations of the data set, as Figure 3.1 shows the complete MLT coverage and Figure 3.2 shows the latitude range of the boundaries.

Each boundary exists in a relatively confined area, and Figure 3.2 highlights the regions where the boundaries are most likely to occur. The PCB extends equatorward to 67^◦ and poleward of 87^◦, with the majority falling between 69^◦ and 80^◦. The IB is present between 57^◦ and 73^◦, with high density between 65^◦ and 70^◦, matching previous IB studies by Sergeev and Gvozdevsky [1995], Donovan et al. [2003a,b], and Lvova et al. [2005]. Finally, the I0 can reach up to 70^◦ and down to 50^◦, with the bulk lying between 57^◦ and 67^◦. The IB is the most concentrated boundary of the three,

Figure 3.1: All three boundaries for all collected FAST orbits, plotted on MLT co-ordinates. The IB (black) and I₀ (red) are closer, on average, than the IB and PCB (blue). Note the equatorward dip in each boundary near midnight (00 MLT).

Figure 3.2: Histograms of the PCB (blue), IB (grey), and I₀ (red), plotted as a function of their magnetic latitude. The sharpness of the IB distribution indicates that it is contained in a smaller latitude range than the other two boundaries. Each histogram contains 10,679 measurements, one for each FAST orbit in the database.

ranging 16^◦ compared to the 20^◦ and above range of the other boundaries.

There is a considerable amount of overlap in latitude for all three boundaries (Fig-ure 3.2), however, recall that there is a definite ordering of auroral boundaries (based on their definition). The I₀ is always equatorward of the IB, which is always equa-torward of the PCB. The overlap is due to the dynamical nature of these boundaries as the whole system can shift up or down in latitude, as it is dependent, in part, on the magnetospheric state. The IB and I₀ are, on average, much closer together than the IB and PCB, seen in Figure 3.2 as the overlapping regions in the boundary distributions. The IB and I₀ share upwards of 12^◦ of overlap compared to the 6^◦ between the IB and PCB.

This chapter will investigate the trends in each boundary as well as their depen-dence on MLT and magnetospheric state. The relative positions of the boundaries will be explored, and the results of some correlation studies will be presented.

3.1 MLT Breakdown

The large amount of data collected can be separated for investigation into the dependence of boundary positions on MLT. The night side was split into six sections of 2 hours of MLT (seen as the pie shaped wedges in Figure 3.1), and the orbits were binned according to the location of their recorded IBs. The breakdown of orbits into each MLT bin can be found in Table 3.1. The resulting binning shows fewer orbits at the dawn/dusk edges, and increasing towards the midnight region.

The dependence of the IB location on MLT has been well documented, so splitting the boundary data based on MLT is reasonable [Lvova et al., 2005]. This allows investigation into the MLT dependence of the I₀ and the PCB, as well as the relative positions of these boundaries. The distributions of boundary positions are shown in Figure 3.3, where the red represents the I₀, grey the IB, and the PCB is marked in

MLT Section No. of Orbits

Table 3.1: Breakdown of FAST orbits by MLT.

blue. Each boundary is represented by a histogram, and each panel depicts a different section of MLT. The MLT section is marked by the shaded region in MLT coordinates located on the right side of each panel. The total number of boundaries is given by N .

The equatorward shift in each boundary near midnight is clearly visible in Figure 3.3. Each histogram moves leftward approaching midnight, and then moves right closer to the dusk and dawn. Also seen in Figure 3.3 are larger spreads in the I₀ and PCB histograms near the dawn and dusk sectors, compared to the relatively stable IB distribution. These distributions are useful in visualizing the relative locations of these three auroral boundaries as they occur in the ionosphere.

A more quantitative investigation into the individual boundaries is necessary for creating a complete picture of the behaviour of each boundary, and is presented in the next few sections.

3.2 Isotropy Boundary (IB)

In order to investigate the trends in the location of the isotropy boundary, a normal curve was fit to each IB distribution. This includes the six distributions for each MLT section (Figure 3.3) described in Section 3.1. The normal distribution was determined to be a reasonable model for the IB through the use of normal probability

Figure 3.3: Histograms of the PCB, IB, and I₀, separated by MLT (indicated by the circular plot on the top right of each panel). Each histogram in a single panel has the same number of measurements, given by N.

cumulative frequency [Montgomery, 2004]. If the plotted data lie along a relatively straight line, then the hypothesized distribution is an appropriate model. The results of the linear regression for each normal probability plot are found in Table 3.2. The regression results for each IB distribution show very linear trends, meaning that using a normal curve to represent the IB distributions at any MLT is a reasonable model for the data. The resulting fits are plotted in Figure 3.4.

MLT Section R² Result

Table 3.2: Regression results for the normal probability plots used for determining if a normal distribution is a reasonable model for each IB distribution. A linear correlation indicates that the normal distribution is an appropriate representation of the data.

The statistics for each fit are found in Table 3.3, and include the mean, median, latitude range (as determined by the latitudes within two standard deviations), and the skewness of the distribution. The trends in the distributions are evident as we examine each section of MLT. The mean and median move equatorward near mid-night, and poleward near dusk and dawn, as expected. The latitude range indicates the amount of spread in the distributions, and shows that the distributions are widest near dusk, and the spread becomes constant from midnight through the dawn sector.

This can be seen in Figure 3.4, where the tails in the top two panels extend further than the other panels, and implies that there is more variation in the IB location near dusk than elsewhere in the night sky.

Skewness is an indicator of how the distribution leans about the mean. A zero-valued skewness means that the distribution is completely centered about the mean,

Figure 3.4: Histograms of the IB latitude, separated by MLT, fitted with normal distributions. The mean is represented by the red vertical line. Note the equatorward shift of the IB near 00 MLT, and the narrowing of the distribution from dusk to dawn.

MLT Section Mean Median Range Skewness

Table 3.3: Fit statistics for the IB distributions in Figure 3.4.

tribution is right leaning (the median lies to the right of the mean), and the tail on the left extends further than that on the right, and vice versa for a positive skewness.

In this case, all the skewness values are negative and the medians lie to the right of the means. The magnitude of the skewness is small and the latitude difference between the medians and means are less than 0.2^◦, so the distributions have slightly longer tails on the left. This implies that the poleward boundary of the IB is more well-defined than the equatorward boundary.

The means from Table 3.3 are plotted in Figure 3.5, as a function of MLT. The resulting shape is parabolic and described by Equation 3.1.

IB_mean = 0.098 (M LT₀)²+ 0.078M LT₀+ 65.1^◦ (3.1)

This equation gives us the minimum mean of an IB distribution as 65^◦, occurring at 23.6MLT.

The result in Equation 3.1 describes the mean IB latitude as a function of MLT.

The distinction is subtle, but important. While the IB itself will reach equatorward of the 65^◦ minimum, this result implies that the mean of any IB samples collected in a certain MLT section will fit on the curve described by Equation 3.1.

Figure 3.5: The means from Table 3.3 plotted as a function of MLT. The parabolic shape indicates that the minimum IB latitude occurs at 23.6MLT.

lowing the same procedure as above, the means of the IB distributions are calculated for four separate Kp bins (1, 2, 3, 4 and greater) and are plotted in Figure 3.6. The equations for each fit are given by

KP 1 : IB_mean = 0.098 (M LT₀)²+ 0.039M LT₀+ 66.3^◦

KP 2 : IB_mean = 0.086 (M LT₀)²+ 0.062M LT₀+ 65.4^◦ (3.3) KP 3 : IB_mean = 0.097 (M LT₀)²+ 0.107M LT₀+ 64.4^◦

KP 4+ : IB_mean = 0.101 (M LT₀)²+ 0.143M LT₀+ 63.1^◦

with M LT₀ given by Equation 3.2. The main difference between the four curves is the vertical positioning, meaning that an increase or decrease in Kp causes the means in all MLT sections to shift up or down. The curves shift consistently downward by about 1^◦ for every Kp bin. This result is a good indicator of the correctness of the database, as it agrees with previous studies of the IB [Donovan et al., 2003b].

This analysis provided some information about how the IB distribution changes across MLTs and its dependence on magnetospheric activity. A normal curve is an

Figure 3.6: Means of IB distributions split into Kp bins, as a function of MLT. The parabolic shape is nearly identical to Figure 3.5, with higher Kp values occurring at lower latitudes.

appropriate model for the IB distribution, and the shape of the curves are consistently right leaning across all night side MLTs (the poleward boundary of the IB is more well-defined that the equatorward boundary).

3.3 Inner Edge of the Ion Plasma Sheet (I

)

The six distributions of the I₀ in Figure 3.3 were subjected to the same normal probability plot test as outlined in Section 3.2, and it was determined that a normal distribution was a reasonable description of the data. The resulting fits are plotted in Figure 3.7, and the fit statistics are shown in Table 3.4.

The trends in the data are again evident as we examine each section of MLT. The equatorward motion of the I₀ near midnight is present, while not as pronounced as the IB, and the means and medians differ no more than 0.4^◦. The latitude range (within two standard deviations) shows how the spread changes from dawn to dusk,

Figure 3.7: Histograms of the I0 latitude, separated by MLT, fitted with normal distributions. The mean is represented by the red vertical line.

MLT Section Mean Median Range Skewness

Table 3.4: Fit statistics for the I₀ distributions in Figure 3.7.

shows the distributions squeezing together near midnight).

The negative skewness once again points toward right leaning distributions, and this can clearly be seen in Figure 3.7 as the heavy tails on the left hand side of the distributions. Once again, we can say that the poleward edge of the I₀ distribution is more well defined that the equatorward edge.

Comparing these results to the IB, it is clear that there is more spread in the latitude ranges of the I₀ than the IB. The differences between the latitude range of each distribution can be as large as 8^◦, with a mininimum of 3^◦. The skewness values are consistently higher for the I₀, meaning that the IB has smaller tails and has a more well defined region of occurrence.

Following the procedure from Section 3.2, the means from Table 3.4 are plotted in Figure 3.8 and fit with the curve described by Equation 3.4 (M LT₀ given by Equation 3.2).

I0,mean = 0.047 (M LT0)²− 0.094M LT0+ 61.7^◦ (3.4) Solving this equation gives us the minimum mean for the I₀ distribution as 61.6^◦, occurring at 1MLT.

Comparing Figure 3.8 to Figure 3.5, we can see that the range of the I₀ means is approximately half of the IB. The spread in IB is close to 3^◦ between MLT sections, and the I₀ spread is close to 1.7^◦. It is important to distinguish that this spread is different than that discussed in regards to the previously determined latitude ranges

Figure 3.8: The means from Table 3.4 plotted as a function of MLT. The minimum mean latitude of the I₀ occurs at 1MLT.

in Table 3.4, which show that the location of the I₀ in each MLT section has a larger range of latitude than the IB. The spread in the means shows that, while the individual range of the I₀ is larger than the IB, the average I₀ location is less variable than the IB. Consider the difference between a wide brush stroke painted across the page and a narrow stroke painted in a parabolic shape; the wide stroke is less variable in its motion, but has a wider range of coverage.

Again, the data in Figure 3.8 was split into sections dictated by Kp: 1,2,3 and 4+. The resulting plot is shown in Figure 3.9 and the fitted curves are given by

KP 1 : I_0,mean = 0.042 (M LT₀)²− 0.126M LT₀+ 63.5^◦

KP 2 : I_0,mean = 0.038 (M LT₀)²− 0.060M LT₀+ 61.9^◦ (3.5) KP 3 : I_0,mean = 0.033 (M LT₀)²− 0.129M LT₀+ 60.6^◦

KP 4+ : I_0,mean = 0.026 (M LT₀)²− 0.282M LT₀+ 58.8^◦

The curves are clearly separated in latitude based on the Kp; higher activity pushes the I₀ into lower latitudes, similar to the IB. However, while the IB curves

Figure 3.9: The means of the I0 distributions split into Kp bins, as a function of MLT.

seem to converge near dawn, the I₀ tends to diverge at higher Kp. The separations between curves are larger than those for the IB, reaching close to 2^◦ between Kp bins.

This analysis into the I0 provided some information about its behaviour across all night side MLTs as well as its dependence on the Kp. The I₀ is similar to the IB, but has some important distinctions, mainly that its overall range of motion across MLT is less than the IB (1.7^◦ to 3^◦), but its range of motion within individual MLT sections is considerably larger (latitude ranges of the I₀ are at least 3^◦ larger than the IB latitude ranges).

3.4 Polar Cap Boundary (PCB)

In this section, we apply the analysis completed in Sections 3.2 and 3.3 to the polar cap boundary. A normal distribution was determined to be a reasonable model for the data across each MLT section, with the fit statistics given in Table 3.5 and the fits themselves shown in Figure 3.10. Once again we see clear shifting of the

shifts may be the most pronounced of all three boundaries, with the means spreading

Table 3.5: Fit statistics for the PCB distributions in Figure 3.10.

The means and medians are separated by less than 1^◦, meaning that the spread of data in each MLT section is larger than the IB, but less than the I₀. The range of latitudes that the PCB occurs is less than the I₀ range, but larger than the IB. The latitude ranges (within two standard deviations) agree with this, and the narrowing of the distribution near midnight is present in the PCB as well.

The distributions of the PCB differ from those of the I0 and IB in that the skew-ness is now positive; meaning that the PCB distributions are left leaning and have larger tails on the right hand side. This is also noticeable in the median data - the

The distributions of the PCB differ from those of the I0 and IB in that the skew-ness is now positive; meaning that the PCB distributions are left leaning and have larger tails on the right hand side. This is also noticeable in the median data - the