Creating a Dataset of Phase Space Density
2. Creating a Dataset of Phase Space Density and Identifying Errors
2.0 Introduction
Appendix A demonstrates how to avoid time periods when the HIST electron flux measurements are anomalous. This chapter will also focus on removing unwanted signal from the data. Specifically, we discuss how to transform HIST electron flux measured as a function of energy and position to phase space density as a function of the three adiabatic invariants. The transformation removes signal generated by the “adiabatic effect” also known as the “Dst effect”. This effect causes large changes in electron flux measured at fixed energy and position but cannot account for flux enhancements that occur during the recovery phase of some storms [Kim and Chan, 1997; Li et al., 1997]. Since the goal of this study is to understand storm time flux enhancements, we must remove flux changes due to the adiabatic effect from our data. This chapter describes the adiabatic effect, how it generates large flux changes, and how we remove these changes from our data. The chapter concludes with a discussion of potential errors introduced by the transformation to phase space density. The chapter gives methods for estimating errors and discusses how errors can affect results.
2.1 The adiabatic effect
The adiabatic effect was first introduced by Dessler and Karplus [1961] to explain the relativistic electron flux changes during geomagnetic storms. Surprisingly, high energy relativistic electrons behave differently during geomagnetic storms than low energy electrons. During a storm, relativistic electron flux is correlated with the Dst index while low energy particle flux is anti-correlated [Lyons and Williams, 1976; Williams, 1981]. The flux of low energy electrons in the inner magnetosphere increases during the storm main phase. The enhanced particle flux increases currents that contribute to the decrease in Dst [Lui et al., 1987]. During the storm recovery phase, low energy particle flux decreases and Dst increases back to its pre-storm levels. Typically, relativistic electron flux correlates to Dst in the opposite manner. Both relativistic electron flux and Dst decrease during the storm main phase and increase during the recovery phase. Dessler and Karplus [1961] proposed that the relativistic electron flux changes were caused by electrons moving to conserve all three adiabatic invariants, even the third invariant, in response to changes in Dst.
The theory, commonly referred to as the Dst effect, applies only to high energy electrons because low energy electrons do not conserve the third invariant during storms. The third invariant is conserved when time variations of the magnetic field are slow compared to the drift period of a particle. Electrons conserving the third invariant satisfy the following inequality [Roederer, 1970]:
[pic] (1)
where B is the field magnitude, dB/dt is the time rate of change of the field magnitude, and τdrift is the drift period of the particle. Rapid magnetic field changes occur during the main phase of a storm at a typical rate of ~20nT/hour. Using this value in the above equation shows that at L=7 only particles with drift periods .7 MeV are controlled by adiabatic motion. Assuming no other loss processes, relativistic electrons conserve the third invariant throughout the storm and react adiabatically to changes in Dst.
Relativistic electron flux variations occur in response to Dst changes as follows. As Dst decreases during the main phase of the storm, electrons conserving their third invariant move outward to compensate for the reduction of field magnitude and still enclose the same magnetic flux in a drift orbit. A spacecraft at fixed radial distance then measures the flux of electrons previously present at lower L. The outward moving electrons are now in a weaker magnetic field and to conserve the first adiabatic invariant, μ, their energy must decrease. The distribution of electrons shifts to lower energy as illustrated schematically in Figure 1. A spacecraft measuring flux at fixed energy, E0, and fixed radial distance measures the flux of electrons initially at a lower L value shifted to lower energies. Generally, the outward movement results in a decrease of electron flux at fixed energy during the main phase of the storm. Any change in electron flux due to the adiabatic response is reversible assuming no loss or addition of new electrons. As the magnetic field returns to its pre-storm value the electrons move back to their original position and the fluxes recover to pre-storm levels.
2.2 Calculating phase space density
We can account for adiabatic flux changes by converting HIST flux measurements, [pic], measured as a function of energy (E), position ([pic]), pitch angle (α) and time to phase space density as a function of the adiabatic invariants and time, f(μ,K,L*,t). Liouville’s theorem states that phase space density of electrons with fixed adiabatic invariants is unaffected by slow temporal changes of the magnetic field and thus is insensitive to the Dst effect . Time changes of f(μ,K,L*,t) result only from new sources or losses of electrons. Electrons accelerated by violation of one or more adiabatic invariants constitute a new source of electrons. It is these new sources and acceleration processes that we hope to understand and that will be illuminated in the phase space density dataset.
Converting the data from [pic] to f(u,K,L*,t) requires multiple steps and computer intensive calculations. Briefly, the method restricts measurements to only those electrons with energy and pitch angle corresponding to fixed values of μ, K, and L∗. The transformation requires calculating the three invariants, μ, K, and L* defined in the introduction along each point of the Polar orbit. The method is broken into the four basic steps outlined below and followed with complete descriptions.
Step 1: Find the particle pitch angles corresponding to constant K along the Polar orbit by tracing particle motion in a model magnetic field.
Step2: Find the particle energy corresponding to constant μ at each point of the orbit.
Step 3: Find the phase space density at the pitch angle and energy of constant K and μ by interpolation.
Step 4: Determine L* by calculating the magnetic flux enclosed in the drift path of a particle with pitch angle corresponding to constant K. The electron drift path and the flux enclosed are calculated in a model magnetic field.
2.2.1 Step 1:
Our method of transforming the data begins with the calculation of K. The invariant K depends on the mirror magnetic field and so varies with particle pitch angle. To measure phase space density of electrons at fixed K throughout the magnetosphere means to measure particles with only specific pitch angles at the observation location. To find the pitch angles corresponding to fixed K, we calculate K values for local pitch angles from 5o to 90o at 5o intervals along the orbit. We use the Tsyganenko 96 (T96) field model [Tsyganenko, 1996] and the UNILIB code provided by the Belgian Institute for Space Aeronomy available on the web at to do the calculations. We create a table containing 18 K values at 20 second time steps along the orbit. Interpolation is done between these values to find the pitch angles corresponding to a given fixed K.
The T96 model used in the calculations varies depending on the value of required input parameters. These input parameters are solar wind By, Bz, dynamic pressure, and Dst. We use the omni dataset as input to the model. The model is valid for only a specified range of solar wind and Dst values. These ranges are 0.5 to10 nPa for dynamic pressure, -100 to 20 nT for Dst, -10 to10 nT for IMF By, and –10 to10 nT for IMF Bz. The measured solar wind or Dst values may fall outside these limits especially during the main phase of a storm,. When this happens we use the maximum or minimum allowed value instead of the measured value. The parameters fall outside of the valid model range only 4% of the time.
2.2.2 Step 2:
The second step is to find the particle energy that corresponds to fixed μ. This is done using the equation for μ to solve for the momentum p:.
[pic] (2)
Here μ is chosen to be a fixed value and αk is the pitch angle of fixed K found in step 1. The relativistic momentum can be written in terms of energy as,
p2=(E2+2moc2E)/c2 (3)
where E is kinetic energy and c is the speed of light. We solve for the kinetic energy by combining the two equations. The energy of a particle with fixed μ depends on magnetic field strength. We use the magnetic field strength measured by the MFE instrument [Russell et al., 1995] onboard Polar to determine the energy.
2.2.3 Step 3:
We obtain phase space density from the HIST flux measurements using the following relation, [pic][Schulz and Lanzerotti, 1974]. Phase space density of electrons at fixed Κ and μ is found by determining f of electrons with pitch angle and energy calculated in steps 1 and 2. Because the HIST instrument measures discrete pitch angles and energies interpolation is necessary. We find phase space density measured at the pitch angle calculated in step 1 by fitting the pitch angle distribution of each energy channel using a downhill simplex minimization routine to the following function.
[pic] (4)
This form was used because it fits both butterfly and highly peaked distributions. From the fitted data we obtain phase space density of electrons with pitch angle of constant K, [pic],at 14 discrete energies. We fit the phase space density as a function of energy to an exponential function. The fit is used to find the phase space density of electrons at the energy of constant μ determined in step 2.
2.2.4 Step 4.
Lastly, we calculate the third adiabatic invariant, L*. This computationally intensive calculation requires tracing the drift of an electron around the entire magnetosphere. This drift is slightly dependent on the pitch angle of the particle. We use the pitch angle determined for constant K to do the calculation. As with the second invariant calculations we use the T96 model and the BISA UNILIB code.
2.3 Errors.
Errors in our calculation of phase space density arise from two sources: poor data fits and imperfect magnetic field models. Poor data fits (step 3) occur because our chosen functional forms may not always accurately represent the data. In addition, during low count periods uncertainties in the measurements become large and result in poor fits. Phase space density errors introduced from poor data fits are quantified by the standard error of the fits to the data. Imperfect magnetic field models result in inaccurate estimates of L* and K. Our goal is to understand how incorrect estimates of L* and K affect the calculated phase space density.
2.2.1 Errors of Poor Fits
Step 3 in our method requires that we fit our measured data to functional forms. We quantify the error in each of these fits by the standard error given below
[pic] [pic] (5)
where s is the standard error, n is the number of data points, σ is the standard deviation, dfit is the estimate of the fit, and dmeasured is the measured data.
2.2.1 L* errors.
L* depends inversely on Φ, the total magnetic flux enclosed by the drift orbit of an electron, and so is affected by the global accuracy of the magnetic field model. An imperfect field model will change L* in predictable ways. If the field model underestimates Φ, the calculated L* will be larger than expected. Likewise, if the field model overestimates Φ, L* will be smaller than expected. Therefore, errors in L* simply shift the calculated phase space density radially as shown by the cartoon of Figure 2.
The cartoon depicts three scenarios. In the first scenario, the field model systematically underestimates Φ and the calculated phase space density versus L* profile shifts to larger L*. In the second scenario, the field model systematically overestimates Φ and the calculated phase space density versus L* profile shifts to smaller L*. In the final scenario the model overestimates Φ at small radial distances and underestimates it at large radial distances. This type of error stretches the phase space density versus L* profile. Inaccurate calculation of L* does not change the value of phase space density. The errors only shift the value radially.
We estimate the error by analyzing how L* changes as the field model is modified. Changing the input parameters modifies the field model. Over the range of distances relevant to this study, the L* parameter is mostly controlled by the Dst input to the model. Therefore we test the dependence on the field model by calculating L* for electrons at a range of radial distance and input Dst values. Figure 3 plots L* values calculated for electrons at X=-8-8, Y=0, and Z=0 in magnetic coordinates using Dst values from 20nT to –100 nT at 10nT intervals. Circles mark L* values for electrons with 90o equatorial pitch angle and asterisks mark L* values for electrons with 20o equatorial pitch angle. The colors of the symbols shows the Dst value used as input to the model. Blue symbols mark L* values calculated using the minimum input value of –100 nT and red symbols mark L* values calculated using the maximum input value of 20 nT.
Before discussing the relevance of the plot to calculating errors we describe some expected features of the plot. These features are worth noting because they clarify the meaning of the L* parameter and its dependence on particle position and pitch angle. For example, the plot shows that calculated L* values decrease with decreasing Dst at all radial distances. This feature is expected because the L* parameter gives the radial distance of an electron when all external magnetic fields are turned off leaving only the dipole field. If the magnetic field measured by Dst were turned off, electrons would move radially inward to conserve their third invariant. The more negative the initial Dst perturbation the farther inward an electron moves and the smaller its L* parameter is. The opposite is true for positive Dst perturbations.
Another noticeable feature is that inside of X=4, L* is not highly dependent on Dst. This is expected because at small radial distance the dipole field of the earth dominates over any perturbations measured by Dst.
One final interesting feature is that L* calculated for 90o and 20o pitch angle particles differ systematically. The plot shows that on the dayside, L* of a 90o pitch angle particle is smaller than L* of a 20o pitch angle particle for the same input Dst value. This feature results from the compression of the magnetic field on the dayside and stretching of the tail on the nightside that causes particle motion known as drift shell splitting [Roederer, 1967]. Drift shell splitting predicts that electrons with 90o pitch angle move radially outward in the compressed field region to remain at constant magnetic field strength and conserve the first invariant. The particles with 20o pitch angle, on the other hand, move radially inward to conserve the second invariant. If the dayside magnetic field compression is turned off leaving only the dipole field, the 90o electron moves radially inward and the 20o particle moves outward. Therefore the L* parameter of the 90o particle is smaller. On the nightside of the magnetosphere particles are affected by drift shell splitting in exactly the opposite manner. Here the equatorial field strength decreases. Particles with 90o pitch angle move inward and those with 20o pitch angle move outward. The plot shows L* of 90o particles is larger than that of 20o particles as expected.
We now discuss how to use the information plotted in Figure 3 to estimate errors in L*. Figure 3 shows that the largest range of L* values at any radial position is ΔL*=2.5. As an estimate of the error we assume a range equal to 60% of this maximum. In other words, our final values of L* are given as L*[pic]0.8. The error in phase space density is the minimum and maximum value calculated within this range of L* values.
2.2.2 K errors.
Calculating K requires integration along a magnetic field line. The integral depends on the length of the field line and the strength of the magnetic field along the field line. The K calculation is compromised by an imperfect field model. The phase space density is affected by an inaccurate estimate of K because inaccurate estimates of K will change the pitch angle determined in step 2 and used in step 3 to find phase space density. Figure 4 demonstrates how erroneous estimates of K affect the phase space density calculations. Panel A is a schematic showing phase space density as a function of K. Large K values correspond to small pitch angle values. Therefore, phase space density typically decreases as K increases as shown. If the model overestimates K values, the phase space density trace shifts to the right. The phase space density of a chosen fixed K will be overestimated. Likewise, if the model underestimates K the phase space density trace shifts left. The phase space density of fixed K will be underestimated. K and phase space density will be overestimated when the stretching of the model is too large. Likewise K will be underestimated when the stretching is too small.
With this understanding we estimate a range of probable phase space density values by modifying the field model and analyzing the change in the calculation of pitch angles of constant K. It is not intuitively clear which input parameters will most affect the calculation of K. We modify each input parameter separately to determine which input parameter has the greatest effect.
We begin by describing how different Dst inputs to the model affect the calculation of pitch angles of constant K. To understand the effect we calculate pitch angles of fixed K for electrons with positions Y=0, Z=0 and X=-8-8 for the full range of Dst inputs from 20nT to –100nT at 10 nT intervals. The other four input parameters are kept at the following constant values, Bz=0 nT By=0 nT and dynamic pressure=.5 nT. Figure 5 plots the results for three values of K and shows several notable features. First, the dependence on Dst is most apparent at large radial distance and hardly noticeable in the inner magnetosphere. This feature is expected because the earth’s dipole field dominates the inner magnetosphere. Also, the effect of Dst is most noticeable at small K values. This feature is understood because changes in Dst predominately modify the magnetic field at the equator. Particles with small K reside near the equator and are most likely to be affected. In addition, there are differences between the day and night side calculations. The calculations on the night side vary more with changes in Dst.
Figure 4 demonstrates that the pitch angle corresponding to constant K depends significantly on Dst especially for small K values. The pitch angles on the nightside at X=-7 and K=100 range from 20o when Dst=-120 to 70o when Dst=20. However, Polar spends much of its time off the equator measuring only particles with large K values. The bulk of the analysis in this thesis relies on particles with K>1000. At these K values the maximum range of pitch angles for models with different Dst input values is ~20 degrees.
Next we look at the effect of Bz on the model field and calculation of pitch angles of fixed K. Figure 5 shows the effect of Bz on the pitch angles corresponding to constant K and is plotted using the same format as Figure 4 discussed above. For these calculations we vary the Bz input value from –10nT to 10nT while holding the other input variables constant at By=0 nT, Dst=0 and dynamic pressure=0.5 nPa. Comparing Figure 5 to Figure 4 reveals that the pitch angles are less dependent on the Bz input than the Dst input. The maximum range of pitch angles at a given L* in Figure 5 is only 8o.
Varying By has an even smaller effect on the calculation of pitch angles of fixed K as shown by Figure 6. In this figure By varies from –10 to 10 nT at 2 nT intervals while the other input parameters are held constant at Bz=0, Dst=0 and dynamic pressure=0.5 nPa. Varying By only changes the pitch angle by 2o.
Lastly we analyze changes in the pitch angle corresponding to fixed K relative to variations of dynamic pressure. Figure 7 plots the dependence of the pitch angle on dynamic pressure. In this figure dynamic pressure varies from 0.5-10 nPa. The maximum range of pitch angle calculated using different input dynamic pressures is 10o. The change is less than that observed when varying Dst. It is interesting to note that increasing the dynamic pressure affects the day and night side pitch angles of constant K in an opposite manner. On the dayside, increasing the dynamic pressure causes the pitch angle of constant K to also increase. On the nightside, increasing dynamic pressure causes the pitch angle of constant K to decrease. The differences are explained as follows. A crude approximation of K is K=[Bmirror-Beq]1/2 s where Bmirror is the mirror point magnetic field, Beq is the equatorial magnetic field and s is the field line length between mirror points. Changing dynamic pressure changes the field magnitude. Assuming the field strength changes nearly uniformly along the field line yields
Kcompressed=C[Bmirror-Beq]1/2 s (6)
where C is the compression factor. However, changing the field uniformly does not change particle pitch angles, αeq=asin(Bmirror/Beq). Therefore, particles with the same equatorial pitch angles now have Kcompressed=CKuncompressed. On the dayside, the compressed field increases and C>1. For these particles Kcompressed is greater than Kuncompressed. Therefore the pitch angle of constant K also increases. On the nightside, compressing the field has the opposite affect. In this region the field decreases, C ................
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