Theoretical predictions for chorus frequency sweep rates by Helliwell and Trakhtengerts are compared with observations from the THEMIS satellites and a previously published dataset from the Cluster satellites. We first extend the theories to use a general magnetic field model to include the effects of magnetic local time and geomagnetic activity, and then show that both theories give the same dependence of the frequency sweep rate on background plasma parameters. The theoretical scaling of frequency sweep rates are shown to agree very well with observations. We demonstrate that for a given equatorial magnetic field strength, nightside and dawnside chorus waves have higher frequency sweep rates because of the stretching of the magnetic field, while dayside chorus waves have lower frequency sweep rates because of the compression of the field. Increasing geomagnetic activity will enhance the asymmetry by increasing the day-night asymmetry of the background field. The results are important for understanding the generation mechanism of chorus waves.
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 As an important part of understanding its generation mechanism, various theories have been suggested to explain the frequency variation of chorus. Helliwell  discussed a phenomenological theory for discrete Very Low Frequency emissions where the change of wave frequency f is caused by the inhomogeneity of the background field. Trakhtengerts  and Trakhtengerts et al.  considered the magnetosphere as a cyclotron maser (MCM) and proposed that chorus waves are generated in the backward wave oscillator (BWO) regime of the MCM. Both Helliwell  and Trakhtengerts  qualitatively estimated the frequency sweep rate Γ ≡ ∂f/∂t as a function of background plasma parameters such as the electron density and the dipole magnetic field. On the other hand, a recent numerical simulation and theory by Omura et al.  suggested that the generation of chorus is caused by the resonant current formed by the electromagnetic phase space hole and the frequency sweep rate of chorus is directly related to the wave amplitude.
 Recently, these theoretical frequency sweep rates have begun to be tested using satellite observations. For example, Macúsová et al.  used observations from the Cluster satellites and demonstrated the dependence of Γ on electron density ne as Γ ∝ neb with b = − 0.44 ± 0.18 or − 0.46 ± 0.17, where Γ is in kHz/s and ne is in cm−3, depending on the slightly different input parameters used. They concluded that the scaling index b is close to the theoretical value of −2/3 from the BWO model by Trakhtengerts  within the estimated experimental error. Another verification of the BWO model has been done by Titova et al.  using MAGION 5 satellite data. Cully et al. , on the other hand, used observations by THEMIS and demonstrated the relationship between Γ and wave amplitude predicted by Omura et al. .
Helliwell  and Trakhtengerts  estimated the frequency sweep rate in a dipole magnetic field, but it is straightforward to extend their results to a more general field where the magnetic field near the equatorial plane can be represented as B(s) = B0(1 + βs2), where B(s) is the magnetic field at a distance s along the field line from the equatorial plane. For a dipole field, the inhomogeneity parameter β = 4.5/r02, where r0 is the equatorial radial distance of the field line. To incorporate the effects of MLT and geomagnetic activity characterized by the Kp index, we use the T89 magnetic field model in this paper and obtain β by fitting B(s) near the equatorial plane as a function of s2 for a given MLT and Kp.
 In the work by Helliwell , the change of frequency is mainly caused by the inhomogeneity of the background field. For the general parabolic field, it is easier to use equation (14) of Helliwell  to calculate the theoretical frequency sweep rate ΓH for a parallel propagating wave as
where fce is the local electron gyrofrequency at s, vg is the wave group velocity, v∥ is the corresponding parallel resonant velocity, and α is the pitch angle. Here and below the refractive index is assumed to be larger than one. We use α = 30°, the working value derived by Helliwell to maximize the transverse current assuming an isotropic distribution and that all resonant electrons are phase-bunched. The dfce/ds in equation (1) is evaluated at a characteristic location lH where the relative phase between the unperturbed electron velocity and the magnetic field of the wave has changed by π. For the general parabolic field given above lH = (v∥/βfce0)1/3 following Helliwell  with fce0 the equatorial electron cyclotron frequency. In our evaluation of ΓH below, we will simply use a representative frequency f = 0.3 fce0for lower-band risers which generally extends from 0.25fce0 to 0.4 fce0 [Burtis and Helliwell, 1976]. Correspondingly, we approximate vg/(1 + vg/v∥) by 0.4v∥, since vg/v∥ = 2f/fce for a parallel propagating wave. Note that with a generalized parabolic field with inhomogeneity parameter β, ΓH is now dependent on MLT and Kp through β.
 It is interesting to compare the frequency sweep rates of Helliwell  and Trakhtengerts . Using the property that the resonance wave number k = 2π∣fce − f∣/v∥, we can show that and for f = 0.3fce0. Here fce0 and β depend on r0, MLT, and Kp. The resonant velocity v∥ = c(fce − f)3/2/fpef1/2 for a parallel propagating wave, where fpe ∝ ne1/2 is the plasma frequency. Thus both theories give the same dependence of Γ on the background plasma parameters, and ΓBWO is smaller than ΓH by roughly a factor of 4, given that we use α = 30° in ΓH and Q = 2 in ΓBWO. Since we are mainly interested in the qualitative dependence of the frequency sweep rate on background plasma parameters, we will only show calculations using ΓH below.
3. Comparison Between Theory and Observation
 We use magnetic field waveform data measured by the Search-Coil Magnetometer (SCM) [Le Contel et al., 2008] from the three inner THEMIS satellites [Angelopoulos, 2008] during the period of 2008/06/01 to 2011/06/01. The Flux-Gate Magnetometer (FGM) [Auster et al., 2008] measures the background magnetic field, which is utilized in this study to calculate local electron gyrofrequencies. The standard deviation of the background magnetic field measurement is insignificant (∼0.1 nT). The total plasma density is inferred from the spacecraft potential and the electron thermal speed following the method of Li et al. , which generally has an uncertainty within a factor of two [Li et al., 2010]. We select time intervals where chorus packets and frequency sweep rates can be clearly identified visually. Each time interval lasts about 6 to 14 seconds, and only lower band risers are considered in this work. We also limit ∣λM∣ ≤ 3°, with λM the magnetic latitude, to minimize the effects of wave propagation on Γ. The wave spectrum of one selected time interval is shown in Figure 1 (bottom). In total, the database includes 149 different time intervals and 1106 chorus packets within them. Our selected chorus waves are mainly distributed on the dayside and dawnside, with r0 between 5 and 9, as shown in Figure 1. Most events occur when the electron densities are less than 16 cm−3, and those having very high ne occur in plasmaspheric plumes. The maximum and the median Kp index for these time intervals are about 4 and 2 respectively, thus all selected events occur in a moderately disturbed period.
 To calculate the frequency sweep rate of observed chorus packets, we manually select a few points along each packet on the time-frequency spectrogram and perform a linear fitting off to time t to obtain the slope ΓOB, as demonstrated by one packet in Figure 1 (bottom). The distribution of ΓOB with respect to the equatorial radial distance r0 is shown in Figure 2 by black dots. The theoretical frequency sweep rates ΓH (light blue dots) in Figure 2 are calculated by equation (1) using the T89 magnetic field model and ne, MLT, Kp index, and r0 from observation. For a given value of α in equation (1), the main uncertainty of calculating ΓH is from the use of T89 magnetic field model and the measurement error of the electron density. While we do not have an estimate of the uncertainty associated with T89 model, the uncertainty of ΓH due to the error of ne can be estimated as δΓH/ΓH = − (2/3)δne/ne. Because δne/ne has a value ranging from −0.5 to 1 [Li et al., 2010], the value of δΓH/ΓHis estimated to be from −0.67 to 0.33. Using a better magnetic field model or multi-point measurement of the background magnetic field [e.g.,Kozelov et al., 2008] might further improve the accuracy of ΓH. The black line in Figure 2 given by y = 17719 r0−5.08is a power-law fitting to ΓOB and the light blue line is y = 8640 r0−5.13, the power-law fitting to ΓH. Dashed lines are the corresponding standard deviation of the results from the fitting function. It could be seen that in general ΓH is smaller than ΓOB by about a factor of two, while the theoretical scaling of the frequency sweep rate with respect to r0 is in good agreement with observation. It should be noted, however, the value of α = 30° derived by Helliwell  might be too small, because it is generally easier to phase bunch electrons with α between roughly 40° and 75° [Inan et al., 1978]. Using a larger α in equation (1) can further improve the agreement between theory and data. For example, using α = 72° brings the average agreement between the fitted value of ΓH and ΓOB within ∼3%. Similarly, the value of Q in the BWO model used above (Q = 2) might also be underestimated; e.g., Titova et al.  suggested that, on average, Q ≈ 8 for lower band chorus calculated using observational frequency sweep rates. However, the exact value of α or Q should be determined through more rigorous numerical simulations and observed distribution functions.
 The chorus frequency sweep rate is also dependent on Kp and MLT as demonstrated previously by Burtis and Helliwell . We show the effects of MLT by comparing ΓOB from dayside (MLT > 8 h) and dawnside (MLT < 8 h) in Figure 3 (left). To minimize the effects of ne, we only select data with ne ≤ 15 cm−3. It could be seen that in general dawnside chorus has a larger frequency sweep rate for a given fce0. According to theories of Helliwell  and Trakhtengerts , both MLT and the geomagnetic activity affect the chorus frequency sweep rate by changing the magnetic field configuration and thus the inhomogeneity parameter β. To demonstrate the effects of MLT and Kp, we show ΓH as a function of MLT for two constant equatorial cyclotron frequencies (fce0 = 2.2 kHz and 3.0 kHz, and ne = 5 cm−3) at three different Kp's in Figure 3 (right). Generally, ΓH is higher on the nightside and dawnside than on the dayside, consistent with observations, because of the compression of the background field on the dayside and the stretching of the field on the nightside. Increasing geomagnetic activity (larger Kp) will increase the asymmetry in ΓH. Also the MLT and Kp effects are more important for larger r0 (fce0 = 2.2 kHz), because the magnetic field of the outer magnetosphere (larger r0) is more strongly affected by geomagnetic activity.
Macúsová et al.  showed the dependence of Γ on electron density ne using observations by the Cluster satellites. The dataset used by Macúsová et al.  has McIlwain parameter LM between 4 and 4.6 and the density ne between 4 cm−3 and 105 cm−3. Most of their data are obtained on the nightside or dawnside. They fit the observed frequency sweep rate ΓM as a function of ne by with the mean value of aM between 18.3 and 19.3 and bM between −0.44 and −0.46, respectively. They compared bM with −2/3 which is the theoretical scaling index if other parameters are kept constant. Here we assume that r0 is roughly equal to LM and use the published data (MLT, Kp, ne, and r0) of risers in Table 2 of Macúsová et al.  to calculate theoretical frequency sweep rate ΓH, shown in Figure 4. The calculated ΓH's are then fitted by and we have aH = 1.5 and bH = − 0.46. Thus by including effects of MLT, Kp, and r0 that were not considered by Macúsová et al. , the agreement between the scaling indices of theoretical frequency sweep rates and of data is greatly improved. The value of aH is again smaller than aM by roughly a factor of two, consistent with the results obtained above using the THEMIS dataset.
 In this work, we showed the calculation of chorus frequency sweep rates using the theories of Helliwell  and Trakhtengerts  with the T89 magnetic field model to include the effects of MLT and geomagnetic activity. By assuming a previously used value for the undetermined parameter Q of Trakhtengerts et al.  and α of Helliwell , we showed that both theories give the same dependence of the frequency sweep rate on ne, fce0 and background magnetic field inhomogeneity parameter β. The estimated sweep rate ΓBWO from Trakhtengerts  is about four times smaller than ΓH from Helliwell , given the value of Q and α used. However, it should be noted that both theories of Helliwell  and Trakhtengerts  were developed to estimate frequency sweep rate only qualitatively. Thus we mainly focused on the qualitative dependence of the frequency sweep rate on background parameters in this work.
 The theoretical predictions were compared with observations from THEMIS and the published dataset of Macúsová et al. . We demonstrated the agreement between theories and observations in terms of the scaling of Γ as a function of r0 and ne by including the effects of MLT and geomagnetic activity that were not considered in previous work. We presented using observations from THEMIS that for a given fce0, the dawnside chorus has larger frequency sweep rates than the dayside chorus, consistent with previous results. We demonstrated that theoretically MLT and the geomagnetic activity affect the sweep rate by changing the background field configuration. For a given fce0, the dayside compression results in a smaller Γ, while the stretching of the background field on the nightside leads to a larger Γ, consistent with observations. This asymmetry in Γ will increase with increasing geomagnetic activity and radial distance, since the magnetic field configuration of the outer magnetosphere is more strongly affected by geomagnetic field activity.
 Various theories [Trakhtengerts, 1995; Omura et al., 2008] have also predicted the relationship between the frequency sweep rate and the wave amplitude. Cully et al.  have verified the relationship of Omura et al. using 21 chorus packets from a 12-minute time interval. While we focus on the dependence of Γ on background plasma parameters in this work, it is important to investigate whether the relationship between Γ and the wave amplitude holds for the larger dataset used in this work where all background parameters vary significantly. We leave this to future work.
 This research was supported at UCLA by NSF grant 0903802, which was awarded through the NSF/DOE Plasma Partnership program, and NASA grants NNX11AR64G and NNX11AD75G. The authors acknowledge O. Le Contel and A. Roux for use of SCM data and K. H. Glassmeier, U. Auster, and W. Baumjohann for the use of FGM data provided under the lead of the Technical University of Braunschweig and with financial support through the German Ministry for Economy and Technology and the German Center for Aviation and Space (DLR) under contract 50 OC 0302. We also thank World Data Center for Geomagnetism, Kyoto for providing the Kp index.
 The Editor thanks Olga Verkhoglyadova and Boris Kozelov for assisting in the evaluation of this paper.