Previous work has established that the linear correlation of the low-altitude particle cusp latitude with the southward component of the IMF is about 0.70. Several possibly better candidate functions for determining the coupling between the magnetosphere and the solar wind have been advanced, but none have been evaluated in terms of the cusp, which is a site of direct solar wind–magnetosphere interaction. On the basis of 11 years of DMSP satellite particle data from 1984–1994 (with verification from the subsequent 11 years, 1995–2005), we find that the best solar wind–magnetosphere coupling function involves electric field dimensions, such as the half wave rectifier (vBs) and the Kan-Lee electric field (EKL = vBTsin2(θc/2), where θc is the IMF clock angle). Both the half wave rectifier (r = 0.77) and the Kan-Lee (r = 0.78) functions have a linear correlation with cusp latitude which is noticeably better than the Bz function used in previous work, and also better than the ɛ parameter (ɛ = vB2sin4(θc/2)). However, the best correlation is with a function whose clock angle dependence is intermediate between the pure half wave rectifier (which implies no merging for Bz > 0) and the Kan-Lee function. Namely, EWAV = vBTsin4(θc/2) correlates with cusp latitude at the r = 0.81 level. This latter clock angle dependence has been previously suggested at various times by J. R. Wygant, by S.-I. Akasofu, and by V. M. Vasyliunas. The improved result holds for both the equatorward and poleward edge of the cusp, and regardless of how the IMF is propagated. Earlier work on cross polar cap potentials and on nightside auroral luminosity also favored the EWAV function, which in combination with our findings suggests a widely applicable result. Dayside merging is thus clearly not purely component driven, as the half wave rectifier formula implies. These results also suggest, albeit less convincingly, that merging shuts down for increasingly northward IMF more rapidly than the Kan-Lee electric field implies.
 The low-altitude particle cusp is the site where the solar wind first couples to the ionosphere, and is perhaps the magnetospheric phenomenon most directly controlled by the solar wind and interplanetary magnetic field (IMF). Historically, the discovery of the low-altitude particle cusp [Heikkila and Winningham, 1971] (hereafter more simply the “cusp”, while the external magnetic indentation on the magnetopause will be termed the “magnetic cusp”) and the realization that its latitude exhibits a clear Bz dependence [Burch, 1972] were among the earliest clear signs of the overarching importance of magnetic merging in driving the magnetosphere.
 Although the central importance of merging to solar wind–magnetosphere coupling is no longer in dispute, work continues to find the best functional form for the coupling between solar wind input parameters and various measures of the magnetospheric response. Probably the candidate coupling functions most often encountered in current space physics research are (1) the half wave rectifier (vBs, which is 0 for northward IMF, and vBz for southward); (2) the ɛ parameter of Perreault and Akasofu  (ɛ = vB2sin4(θc/2)); and (3) the Kan-Lee electric field (EKL = vBT2sin2(θc/2)[Kan and Lee, 1979]. Here the IMF clock angle is defined by θc = arctan(By/Bz). All three consistently produce better results than Bz does alone. Determining which of the various possible coupling functions works best is desirable, both to improve ability to predict space weather, and because of the insights likely to be shed on the theoretical underpinnings of solar wind–magnetospheric coupling. However, to the best of our knowledge, none of these commonly encountered candidate coupling functions have been tested against the cusp latitude, despite the immediacy of its connection to the solar wind.
 Cusp latitude depends on seasonal (dipole tilt angle) variations, IMF variations, and solar wind pressure. By far the highest correlation previously established is with IMF Bz. Essentially everyone who has investigated the question has found the same consistent relationship between the magnitude of Bz < 0 and cusp latitude. Cusp behavior does vary with altitude. The higher the altitude, the greater the influence of solar wind pressure on cusp position – while the effect of the IMF, although still present, moderates somewhat. This is because high-altitude magnetospheric magnetic field lines can be appreciably compressed by the solar wind [e.g., Zhou et al., 2000], while the position of the low-altitude cusp depends on the open/closed boundary – that is, primarily on the amount of erosion of dayside magnetic flux by merging.
Burch [1972, 1973] presented the earliest results. These observations were taken before the existence of the frontside low-latitude boundary layer (LLBL) was established from high-altitude data [Haerendel et al., 1978], and therefore before the distinction between cusp and LLBL precipitation [Newell and Meng, 1988] was clear at low altitude. Thus Burch  reports a cusp thickness of 4°, as opposed to the average of 1° reported by Newell and Meng . Actually the difference is not as crucial to this type of study as might be at first supposed, since both boundaries must move together. Burch found that the cusp shifted southward by about 5° in MLAT as IMF Bz went from 0 to −5 nT. These early results represent a powerful (although underappreciated) early observational proof of the concept of magnetic merging. (The magnitude of the cusp shift in latitude in this early report is somewhat higher than in subsequent reports with larger data sets).
 Similar results were reported by Carbary and Meng . They used electron moment data only from the early generation DMSP satellites to identify a “cusp” position. By using high time resolution IMP 8 observations, limited to intervals within 6 hours of a bow shock crossing, they obtained a formula for the cusp latitude
 Most of the Carbary and Meng  observations were for southward IMF conditions or for small magnitude ∣Bz∣ (although they did not separate northward and southward cases). They did however restrict IMP 8 data to times within 6 hours of a bow shock crossing, so that the IMF impacting the magnetosphere was unusually well determined, at least for statistical work.
Newell et al.  used a more careful definition of the cusp, but made less careful use of IMF data (using hourly averages of IMP 8 data regardless of location with respect to the bow shock). They reported
Equation (3) implies that the cusp position changes relatively little from its nominal position (about 77°–78° MLAT) for increasing magnitude northward IMF. Thus with respect to the cusp latitude, Newell et al.  endorsed the view that the magnetosphere–solar wind coupling acts like the so-called “half wave rectifier” effect. Subsequent studies have found a similar distinction between southward and northward IMF dependencies of cusp latitude [e.g., Zhou et al., 2000; Wing et al., 2001].
Zhou and Russell  used Hawkeye magnetic field data to study the IMF dependencies of the high-altitude magnetic indentation on the magnetopause (which can be called the “magnetic cusp”). Although the magnetic cusp is not necessarily identical to the particle cusp, Zhou and Russell  did find the same qualitative behavior as in these early low-altitude studies. Specifically, they found that the magnetic cusp position moved from an average position of 77° GSM for Bz > 0 to a latitude of 74° for Bz < 0.
 Finally, Zhou et al.  used Polar magnetic field and particle data to identify the cusp with good precision at high altitude but inside the magnetosphere. They found that the average cusp latitude was 80.3° invariant latitude, but shifted equatorward for southward IMF, reaching 73° invariant latitude for Bz = −10 nT. The magnitude of the slope for cusp latitude-IMF Bz dependence is thus about 0.7°/nT, in fairly good agreement with the results of Newell et al. .
 To date then, studies of the solar wind control of the cusp have focused almost exclusively on a single simple driving function, namely, Bz (or Bs, which is zero for northward IMF, and Bz for southward). Meanwhile, a great many additional candidate coupling functions have been proposed by various theorists, while experimenters have made a variety of empirical fits for the magnetospheric response as the cross polar cap potential, Kp, AE, Dst, and nightside auroral luminosity [Crooker et al., 1977; Wygant et al., 1983; Liou et al., 1998; Papitashvili et al., 2000]. We are here interested in those candidate coupling functions which have both theoretical support, and success as tested elsewhere against other geophysical parameters. Coupling functions which are not relatively simple, or which appear to be ad hoc fits to a very specific database are not of interest here. Thus the aim is to find not only a better predictor of cusp latitude than Bz, but also to find a predictor which is likely to be widely applicable in space weather forecasting. Experience teaches that complex formulations rarely have satisfactory adaptability when applied to new data sets.
 Two previous studies testing candidate functions are particular relevant to the present work. Wygant et al.  tested coupling functions which were then, and still are, popular. They concluded that the cross polar cap potential was best represented by an “intermediate” function, EWAV = vBTsin4(θc/2). Wygant et al. termed this function “intermediate” because its response to the IMF clock angle is intermediate between the half wave rectifier, and the Kan-Lee electric field. In fact, the functional dependence on clock angle is the same as that discussed by Vasyliunas et al. , and the same as used in the ɛ function of Perreault and Akasofu . Likewise, Liou et al.  evaluated several solar wind predictors of the global auroral luminosity, as inferred from Polar UVI measurements, and also found that the “intermediate” formula, which we hereafter term EWAV (for “Wygant”, “Akasofu”, and “Vasyliunas”), worked best.
2. Data and Techniques
2.1. DMSP Satellites and the SSJ/4 Instrument
 Data from the SSJ/4 electrostatic analyzers on the DMSP series satellites (F6 through F12, and a small amount of F13) from 1 January 1984 through 31 December 1994 were used to compile cusp boundaries. These 11 years, besides covering all phases of a solar cycle, in general contain higher-quality data than some later years, during which increased operational longevity produced deterioration of sensitivity, and hence the reliability of boundary identification in several detectors. During recent years, some SSJ/4 detectors have been launched with the low-energy ion head performing suboptimally, which has also reduced the quality of the ion precipitation data. For these reasons, our studied concentrated on an even 11 years (one solar cycle) worth of good quality data. Nonetheless, we did repeat the calculations for the subsequent 11 years, to test whether the relative rankings of the coupling functions changed (as discussed in section 3.3, they did not).
 The DMSP satellites are in Sun-synchronous nearly circular polar orbits at about 845 km altitude, with orbital inclinations of 98.7°. The orbits of the DMSP satellites are such that the least covered regions are postnoon and especially postmidnight, except at high magnetic latitudes. The SSJ/4 instrumental package included on all these flights uses curved plate electrostatic analyzers to measure electrons and ions with one complete spectrum each obtained per second [Hardy et al., 1984]. The satellites are three-axis stabilized; and the detector apertures always point toward local zenith. At the latitudes of interest in this paper, this means that only highly field-aligned particles well within the atmospheric loss clock are observed.
2.2. Background on Cusp Plasma Characteristics
 A series of articles researching the quantitative differences between the various dayside precipitation regions has been previously presented. For example, cusp identification [Newell and Meng, 1988], mantle identification [Newell et al., 1991a] and LLBL identification [Newell et al., 1991b; Newell and Meng, 1998] have all been the subject of dedicated research papers. Although the shocked solar wind is the ultimate source for much of the dayside precipitation at higher latitudes, clear quantitative differences between these precipitation regions have been documented in the literature.
 The cusp (more specifically, the “low-altitude particle cusp”) is the region where precipitating particle energies and densities approximate that of the frontside magnetosheath. Ions have spectral flux peaks somewhere around 1 keV, reflecting their original solar wind kinetic energy, with spectral flux peaks of about 108 eV/cm2 s sr eV (or number flux peak of 105/cm2 s sr eV). Electrons have temperatures of a few tens of eV, and densities which closely match the ions (usually in the 1–10/cm3 range). The particle cusp thus includes field lines merged from about 1 to 3 min ago, long enough for the main ion population to reach the ionosphere, until about 10 min old, when the high-altitude field line has convected sufficiently far downstream that ion entry is limited.
2.3. Cusp Identification Algorithm
 We identified the cusp using an algorithm quite similar to that of Newell and Meng . That paper showed that a variety of traits characteristic of the magnetopause plasma could be observed clustered together at low altitude. As in the work by Newell and Meng , a two-stage process was used, with each individual spectrum evaluated according to these criteria: (1) iave ≤ 3000 eV; (2) eave ≤ 220 eV; ion spectral flux peak ≥ 2.0 × 107 eV/cm2 s sr; (3) the ion spectral flux peak occurred between 100 eV and 7000 eV. Here iave (eave) refers to the moment calculation of ion (electron) average energy.
 To avoid reader confusion, it is worth noting that the average energy of precipitation (2kT) is higher than the average energy of a plasma population (3kT/2). Moreover the finite lower energy bound (32 eV) of the SSJ/4 instrument raises the electron average energy as computed by moments. A Maxwellian fitted temperature to the cusp electron population is thus typically in the tens of eV range. However, fitting Maxwellians is subject to additional problems, and does not improve the accuracy of the cusp identification in the DMSP data set.
 During the second stage or pass through the data, a sliding window of 4 s (4 spectra) was used. The cusp was deemed to have been entered when 3 out of 4 (1 s) spectra fit the cusp criteria. The cusp was deemed to have been exited when 3 out of 4 spectra did not fit the cusp (more precisely, the boundary was placed at the last spectrum identified as cusp after 3 spectra out of 4 were identified as not cusp).
2.4. IMP 8 IMF Data
 We downloaded the IMP 8 magnetic field and plasma data from the National Space Sciences Data Center (NSSDC) at NASA Goddard, at both the highest time resolution available (about 15 s for magnetic field data and 1 min for plasma data) and at hourly average resolution. There are several possible ways to correlate the IMF with the cusp latitude. Interestingly, none of the ways we tried changed the relative order of goodness of the various coupling functions tested, and the precise procedure chosen for handling the IMF actually produced only modest differences in the correlations. We used the Weimer et al.  code to propagate the high time resolution IMF data and found the best results using a 60 min integration time (ending at the time of contact with the magnetopause).
 However slightly better results were obtained using the hourly IMF averages compiled by NSSDC, which are not propagated, although IMP 8 is generally fairly close to the magnetopause. The best results came from using data from the current hour (i.e., the epoch of the cusp measurement) and the hour preceding the cusp measurement, accorded relative weights w and 1-w, where w is fraction of the hour expired at the time DMSP encountered the cusp. Thus at the beginning of an hour, the previous hour is heavily weighted, while toward the end of an hour, the current hour is heavily weighted. This simple approach modestly outdid the other variations we tried.
 Note that the cusp latitude does represent a time integration over both dayside merging, and the return of flux from nightside merging. Thus it is not surprising that an integration that involved both the current hour and the previous hourly value worked well. Attempts to include still earlier times, even with very low weights, only reduced the correlations. Thus the cusp latitude seems to depend roughly on the last 60–120 min of IMF input.
 An additional advantage of not propagating the IMF is faster computer runs through the whole data set. Since the IMF treatment did not affect the main focus of this paper, the relative ability of various coupling functions to predict cusp latitude, we will not dwell further on the minor variations produced by major variations in the handling of the IMF data.
3. Cusp Dependencies on Various Coupling Functions
 For the purposes of this study, the cusp boundaries identified by the algorithm outlined in section 2 were passed through certain further constraints. These were (1) the MLTs of both equatorward and poleward boundaries were greater than or equal to 11 MLT, and also both less than or equal to 13 MLT; (2) the satellite data had no data gap down to least 65.0 MLAT; and (3) the satellite pass included data up to at least 78.0 MLAT. The latter conditions are to preclude highly slanted orbits from identifying artificially low or high boundaries to the cusp. Altogether 1857 cusp equatorward (and poleward) boundaries were selected after these limitations.
 IMF hourly data were available for only about one third of the cases. Therefore only 499 cusp passes were available for use in investigating the coupling functions over our main period of study, 1984–1994. (Incidentally, the verification interval, 1995–2005, had n = 4835, reflecting greatly improved IMF data availability.)
3.1. Cusp Latitudinal Dependence on Various Coupling Functions
 Altogether we considered 10 coupling functions frequently encountered in the literature, 9 of which are shown in Table 1 (the omitted one is an additional formula proposed by Vasyliunas et al.  which performed no better than Bz alone). Table 1 gives their correlation with the cusp equatorward boundary, cusp poleward boundary, cusp width, and with the cusp equatorward boundary after correction for dipole tilt effects. Since correcting for dipole tilt does not affect the order of merit among the coupling functions, but does improve the correlations, we will present plots for just the dipole-corrected equatorward boundary (section 4 gives more on dipole tilt).
Table 1. Correlations Between Various Proposed Solar Wind Input Functions and the Cusp Equatorward Boundary, Poleward Boundary, the Corrected Cusp Equatorward Boundary (With Dipole Tilt Dependence Removed), and Cusp Widtha
Equatorward Cusp Boundary
Pole Cusp Boundary
Corrected Equatorward Boundary
The optimal input function (WAV) uses the clock angle dependence proposed, at various times, by J. R. Wygant, S.-I. Akasofu, and V. M. Vasyliunas, with the vBT term proposed, among others, by Kan-Lee, J. R. Wygant, and V. M. Vasyliunas. The slope and intercept refer to the fit with cusp equatorward boundary corrected for dipole tilt. The units are v km/s; n cm−3; B nT; p nPa; cusp latitude is MLAT (degrees). Bold values are the highest correlation found, one for each dependent variable.
Figures 1 and 2show the dependence of that boundary upon IMF Bz, for southward and northward, respectively, IMF. To compare with previous work, Figure 1 (unlike Table 1) separately breaks out southward and northward IMF, and thereby improves the correlation. These results verify that the current database and techniques can reproduce previous research. As expected, there is a fairly high correlation (r = −0.69) for southward IMF, and a fairly weak correlation (r = 0.11) for northward IMF. It is easy to see in Figures 1 and 2 support for the notion of the solar wind input as a “half wave rectifier.”
 The three best correlations we found were with the half wave rectifier, the Kan-Lee electric field, and with the formula intermediate between those two, EWAV. The formula by V. M. Vasyliunas adds a p1/6v1/3 factor to EWAV, based entirely on dimensional speculation, but behaves similarly to the latter. This equivalence might be expected, since, for example, even a 100% increase in pressure makes only a 12% difference. Because the Vasyliunas formula is slightly more complex than EWAV, yet produces similar (and for the cusp at least, slightly worse) results, it seems more sensible to use EWAV. However, we do not regard the observational evidence as strong enough to eliminate the original Vasyliunas formula as a candidate for the ultimate “correct” choice.
Figures 3– 5 show these three superior fits. The half wave rectifier (Figure 3) and the Kan-Lee electric field (Figure 4) correlate at the 0.77 and 0.78 levels, respectively, clearly better than Bz (or Bs). The functional form “intermediate” between them, EWAV, correlates (Figure 5) at the 0.81 level. Thus, while Bs explains slightly less than half the variance in cusp position (49%), EWAV can explain 65%, while the half wave rectifier and Kan-Lee functions explain 59% and 61%, respectively. EWAV can account for 66% of the variance in cusp latitude.
 A plot (not shown) of any of these three functions versus a binned mean value of solar wind parameters shows an apparent small nonlinearity. Because nonlinear functions have not been much advocated in the literature, and work only modestly better, we will not discuss them extensively here. It does seem worth briefly noting, however, the best correlation we could find, with free experimentation over many possible (and unpublished) functions, was EWAV2/3, which has r = −0.83, with Λc = −3.65 × 10−2EWAV2/3 + 77.2° (for all coupling functions, the computational units of v are km/s, and of B, nT).
3.2. Cusp Width Dependencies of Various Coupling Functions
 There have been a few reports on the IMF dependence of cusp width, but none of the reported dependencies is very significant. Newell and Meng  reported that the cusp is narrower for southward IMF than for northward IMF, a result which is consistent with the positive correlation coefficient shown in Table 1, although the actual size of the correlation is low enough that it does not seem noteworthy. Zhou et al.  reported a very similar result from high-altitude IMF data. It seems likely therefore that the cusp is slightly narrower for southward IMF, but that the magnitude of the effect is relatively small.
Wing et al.  found that the cusp width is slightly larger for large magnitude IMF By. Although perhaps interesting in the context of the double cusp (which tends to occur for large magnitude By), the absolute value of the effect is quite small, on the order of 0.1° MLAT for 10 nT IMF By.
 The highest correlation seen between IMF inputs and cusp widths in the present research was with solar wind dynamic pressure. This latter correlates with cusp width at the r = 0.31 level. When binned into discrete pressure bins, the effect on the mean cusp width is at any rate consistent, as Figure 6 shows. The effect also holds true over a second solar cycle (1995–2005). There is, therefore, reason to believe the pressure (or density, which correlates at the r = 0.29 level) effect is real, although it also is not particularly large. Since density and pressure are so highly correlated, it would be difficult to be certain which is actually the key solar wind input for affecting cusp width.
3.3. Verification of the Results Over the Subsequent Solar Cycle (1995–2005)
 As mentioned in section 2, the low-energy ion heads on several of the DMSP detectors after 1995 were degraded, so that the automated identifications are not as reliable. Moreover, the IMF data in that period are not from a single source near Earth but from multiple platforms (notably WIND and ACE) located much farther upstream, introducing uncertainty into the time lag required (and even to some extent whether the measured IMF ever encounters the Earth). Nonetheless, NSSDC has provided time-lagged IMF data from this time period, and we do have automated cusp identifications available. We therefore repeated all calculations over the period 1995–2005, hoping to find similar, albeit somewhat degraded results.
 Because IMF data are continuously available, and because more DMSP satellites are operational, many more combined cusp-IMF data points exist for 1995–2005 (n = 4835) than for 1984–1994 (n = 499). The correlations do in fact drop during the later 11 years, but by far less than might have been expected. The most important point is that the ranking of the coupling functions is unchanged from Table 1. Interestingly, even the slopes and intercepts are very similar. Thus the results presented elsewhere in this paper from 1984–1994 apply to other times as well.
 Since the correlations are slightly degraded, we will not report the full set of numbers for 1995–2005. As an example, consider the equatorward boundary of the cusp, corrected for dipole tilt. The highest correlation is with Ewav (r = −0.784) followed by EKL (r = −0.770) and vBs (r = 0.744). The best least squares fit for the corrected cusp equatorward boundary is Λc = 2.4 × 10−3EWAV + 78.4° MLAT. These numbers are very similar to those reported in Table 1 for 1984–1994.
 The high similarity between the results from 1984–1994 and 1995–2005, and the high correlation coefficients over the second 11 years, is only possible if nothing went very wrong over the second 11 years. Thus the particle identifications in 1995–2005, and NSSDC's solar wind lagging from ACE and WIND, must both be reasonably good.
4. Dipole Tilt Dependence
 Early researchers into the Chapman-Ferraro problem, which consists largely of finding a set of magnetopause currents which balances the solar wind pressure, soon found that the location of the magnetic cusp depended significantly upon the angle of inclination between the Earth's dipole axis and the Sun [Mead and Fairfield, 1975; Choe and Beard, 1974]. Early empirical magnetic field models also showed the same dipole tilt angle variation in the cusp latitude. The first test of these predictions based on low-altitude cusp observations was by Burch . The early models got the right qualitative behavior for the cusp, but consistently predicted a cusp latitude higher than is observed at low altitude. Tsyganenko and Stern  introduced the first magnetic field model with the correct low-altitude cusp latitude, largely because that model explicitly includes the dayside Birkeland field-aligned currents. The largest empirical study of the dependence of cusp latitude on dipole tilt angle is by Newell et al. . They reported that for every 1° of dipole tilt, the cusp latitude shifts by 0.06°. The direction of the shift is such that the winter hemisphere cusp is at lower latitudes, as the early magnetic field models predicted.
 In the present study, which roughly doubles the number of cases as that of Newell et al. , we find a moderately smaller slope, namely, −0.043° cusp latitude per 1° dipole tilt. In calculating this number, we made no use of IMF data, which thereby allowed for a much larger number of cusp boundaries, ntilt = 1857, mostly from the southern hemisphere.
 We also checked whether any difference existed between the two hemispheres in terms of dipole tilt angle effects (which might happen, if say, orbital variations such as the Earth being closer to the Sun in January actually constituted part of the apparent dipole tilt effect). Since most of the cusp data are from the southern hemisphere, the latter naturally has the same dipole tilt dependence as the data set as a whole. The smaller subset of data from the northern hemisphere (407 cases) has a dipole tilt slope dependence of −0.046° cusp latitude per 1° dipole tilt. Therefore both hemispheres act substantially similar in this regard.
 Overall, the effect of dipole tilt on cusp latitude is much smaller than the effect of the IMF. For example, the correlation between cusp latitude and dipole tilt is only r = −0.23. Nonetheless, correcting for dipole tilt effects does better the correlations between the coupling functions and cusp latitude. As Table 1 shows, all the coupling functions improve, and all by about the same amount, after correcting for dipole tilt. We did find that a slight apparent under correction for dipole tilt (adding only 0.03° MLAT/1° dipole tilt) produced slightly better results than the full correction. This implies that a small portion of the apparent dipole tilt correction (about 0.01° MLAT/1° dipole tilt) actually owes to a generalized Russell-McPherron effect, in which, for example, the magnitude of the IMF depends on the Earth's noncircular orbit [Newell et al., 2002].
 The key point is that including dipole tilt corrections does improve all solar wind correlations, but does not change the relative order of goodness of any of the coupling functions.
Figure 7 helps clarify the relationship between the three solar wind coupling functions (half wave rectifier, EKL, and EWAV) which best predict cusp latitude in our work, and thus presumably, the extent of dayside merging. All three functions involve vBT times a function of the IMF clock angle. In all three cases, the greatest amount of merging is predicted for a due southward IMF, and all three functions go to zero for a strictly northward IMF. The half wave rectifier model predicts no merging at all for an IMF with any northward component, and thus goes to zero at a 90° clock angle. The EKL function allows considerable merging even past 90° clock angle, and predicts the greatest amount of northward IMF interaction between the magnetosphere and solar wind. The EWAV function lies intermediate between the other two, allowing some merging for northward IMF, but not so much as EKL.
 There is ample evidence in the literature that merging does not in fact go to zero precisely at a 90° clock angle. For example, Gosling et al.  used ISEE 2 data to study the occurrence of flow reversals, indicating merging events, at low latitude (but high altitude) on the frontside magnetopause. They found that although the probability of merging was highest for southward IMF, some merging occurs for local shear angle as low as 40°. The local shear angle is not always the same as the IMF clock angle, of course, but for the low-latitude frontside magnetopause they should be quite similar. We have taken the liberty of replotting the data of Gosling et al.  onto Figure 7.
 When observations suffer from poor counting statistics (here just 17 merging events were found), smoothing may improve the results. This is particularly true when one of several conditions apply. In this case, there is strong theoretical reason to expect that merging is a monotonic function of magnetic shear (i.e., that a smaller shear does not lead to a higher merging rate). Thus we smoothed the data of Gosling et al.  by a nearest-neighbor algorithm (fn = 0.25*fn−1 + 0.5*fn + 0.25*fn+1). The ISEE 2 data are suggestive of EKL, although lying between EWAV and EKL. The small amount of data in the Gosling et al.  study precludes selecting a definite function, except that the half wave rectifier function does seem to be excluded. For merging at low latitudes on the frontside magnetopause, EKL may be a good candidate function.
 Of course merging is known to occur not just at low latitudes on the frontside, but everywhere along the magnetopause. There is good evidence that larger magnetic shears are required for merging to occur away from the low-latitude frontside (which is probably favored for several reasons, including the stagnant magnetosheath flow velocities). In an earlier ISEE study, Gosling et al.  produced a histogram of merging events (the histogram in Gosling et al.'s Figure 10 seems to have 28 events, although the text refers to 29) as a function of magnetic shear as observed at the magnetopause near the dusk flank (i.e., the near-tail magnetopause). While the small frontside study is suggestive of EKL, the flank data suggest the half wave rectifier. The global response of the magnetopause to the IMF must be some combination of low-latitude frontside merging (which allows merging for small clock angles) and merging elsewhere on the magnetopause (which requires larger shears). It is not clear what combination is appropriate. If we naively simply average the behavior near the nose of the magnetopause (the Gosling et al.  study) with that away from the frontside (the Gosling et al.  study), that is, add the two (normalized) distributions together and divide by two, we get the results shown in Figure 8. The combination of merging near the frontside and away from the frontside collectively follows the EWAV function.
 We do not take the shape of the high-altitude response versus shear angle as anything more than an intriguing indication of what might be done with a more extensive data set. In principle, if the frequency of merging as a function of magnetic shear were sampled at several locations along the magnetopause, the data could be combined with a draping model for the magnetosheath magnetic field, and the combination would give the global merging response with enough accuracy to be widely useful.
 Nonetheless, we can be relatively sure that the half wave rectifier function is too restrictive. Considerable additional evidence exists from both high-altitude and ionospheric observations that dayside merging continues for northward IMF. For example, Fuselier et al.  showed that frontside merging can occur for magnetic shears substantially smaller than 90°. Likewise, the polar cap for northward IMF conditions, but with ∣By∣ > Bz > 0 often resembles southward IMF [Newell et al., 1997; Sotirelis et al., 1997; Papitashvili and Rich, 2002].
 One can imagine other “intermediate” functions than EWAV. It would certainly be premature to say that the correct coupling has been is precisely captured. Nonetheless, broadly speaking, it appears reasonable to conclude that, among simple and widely used formulas with theoretical support, the best formulas involve the factor vBT times a function of the IMF clock angle that is less restrictive than the half wave rectifier.
 The latitude of the low-altitude particle cusp is unique in providing a single-point measurement which nonetheless characterizes the global solar wind–magnetosphere interaction in perhaps its most important aspect, the degree to which dayside magnetic field lines have merged with the IMF. All the commonly used measures of solar wind input correlate reasonably well with cusp latitude. Bz correlates less well (r = 0.71) than other popular measures, none of which have been previously tested against cusp latitude. The best functions which have both theoretical support and widespread usage are the half wave rectifier, EKL, and EWAV. Note that all three functions involve vBT multiplied by a function of the IMF clock angle. The former two correlate with cusp latitude at the 0.77 and 0.78 level, respectively, while EWAV correlates at the 0.81 level. Thus cusp latitude joins cross polar cap potential [Wygant et al., 1983] and global auroral luminosity [Liou et al., 1998] in favoring EWAV.
 Because of the relative paucity of high-altitude data (it is much more difficult to sample on a global basis) it is difficult to compare the latter with ionospheric data. However, as the discussion in section 5 shows, the high-altitude data taken near the frontside at low latitude are suggestive of EKL, while merging on a global basis might be roughly consistent with EWAV.
 One result of the work presented here is that the cusp latitude can now be predicted with two thirds of the variance accounted for (r2 = 0.65), as opposed to only half using the previously available Bz formula (r2 = 0.49, which in fact worked only for southward IMF). The broader implications are that the global rate of merging for IMF clock angles below 90° is clearly greater than that predicted by the half wave rectifier model. It is possible that merging on a global basis is reasonably well approximated by EWAV.
 This work was supported by NASA grant NNG05GJ90G to the Johns Hopkins University Applied Physics Laboratory. Dave Hardy of AFRL designed and built the DMSP SSJ/4 particle detectors. Work at AFRL is supported by the Air Force Office of Scientific Research Task 2311SDA3. This research would not be possible without the work of the NSSDC at NASA Goddard, or without the work of scientists such as J. King, R. Lepping, and N. Papatashvilli, who make the solar wind and IMF data widely available in convenient and reliable formats.
 Zuyin Pu thanks Vladimir Papitashvili and Q.-G. Zong for their assistance in evaluating this paper.