The angular momenta of solar system bodies: Implications for asteroid strengths

The angular momentum H is plotted versus mass M for the planets and for all asteroids with known rotation rates and shapes, primarily taken from D. C. McAdoo and J. A. Burns [Icarus18, 285–293 (1973)]. An asteroid's angular momentum is derived from its rotation rate as determined by the period of its lightcurve, its shape as indicated by the lightcurve amplitude, and where possible its size as given by polarimetry or radiometry. The asteroid is assumed to be rotating about its axis of maximum moment of inertia. As previously found by F. F. Fish [Icarus7, 251–256 (1967]) and W. K. Hartmann and S. M. Larson [Icarus7, 257–260 (1967)], H is approximately proportional to $\text{M}{}^{\mathrm{<mtext>5</mtext><mtext>3</mtext>}}$, which shows that the asteroids and most planets spin with nearly the same rate. The very smallest asteroids on the plot deviate from the above reaction, usually containing excess angular momentum. This suggests that collisions have transferred substantial angular momentum to the smallest asteroids, perhaps causing their internal stress states to be substantially modified by centrifugal effects.The forces produced by gravitation are then compared to centrifugal effects for a rotating, triaxial ellipsoid of density 3 g cm^?3. For all asteroids with known properties the gravitational attraction is shown to be larger than the centrifugal acceleration of a particle on the surface: thus the observed asteroid regoliths are gravitationally bound. Poisson's equation for the gravitational potential is investigated and it is shown by mathematical and physical arguments that any arbitrarily shaped ellipsoid with the attractive surface force boundary condition found above will have only attractive internal forces. Thus the internal stress states in asteroids are always compressive so that asteroids could be internally fractured without losing their integrity.     

Coordinated observations of asteroids 1219 Britta and 1972 Yi Xing

Identified as possible flyby targets for the Galileo spacecraft, Asteroids 1219 Britta and 1972 Yi Xing became the focus of a coordinated observing program. Although a subsequent change in the launch date removed these asteroids from consideration for the Galileo mission, the ground-based observing program yielded a substantial amount of information on these previously unobserved asteroids. Britta's sideral rotation period is found to be 5.57497 ± 0.00013 hr and its rotation is retrograde. The lightcurve amplitude ranged from 0.60 to 0.70 mag, depending on phase angle. Britta can be classified as an S-type asteroids based on its measured spectra and albedo. The absolute magnitude and slope parameter derived from the lightcurve maxima are H₀ = 11.67 ± 0.03 and G₀ = 0.03 ± 0.04. A 0.002 mag deg⁻¹ phase reddening in B·V was also measured. 1972 Yi Xing was less well observed but a unique synodic period of 14.183 ± 0.003 hr was determined. The observed lightcurve amplitude was 0.18 mag. Five-color measurements are consistent with an S-type classification. For an assumed slope parameter G = 0.25, Yi Xing's (lightcurve maximum) absolute magnitude H₀ = 13.32 ± 0.01.     

The asteroid lightcurve database

The compilation of a central database for asteroid lightcurve data, i.e., rotation rate and amplitude along with ancillary information such as diameter and albedo (known or estimated), taxonomic class, etc., has been important to statistical studies for several decades. Having such a compilation saves the researcher hours of effort combing through any number of journals, some obvious and some not, to check on prior research. Harris has been compiling such data in the Asteroid Lightcurve Database (LCDB) for more than 25 years with Warner and Pravec assisting the past several years. The main data included in the LCDB are lightcurve rotation periods and amplitudes, color indices, H-G parameters, diameters (actual or derived), basic binary asteroid parameters, and spin axis and shape models. As time permits we are reviewing existing entries to enter data not previously recorded (e.g., phase angle data). As of 2008 December, data for 3741 asteroids based on more than 10650 separate detail records derived from entries in various journals were included in the LCDB. Of those 3741 asteroids, approximately 3100 have data of sufficient quality for statistical analysis, including 7 that have “dual citizenship” - meaning that they have (or had) asteroid designations as well comet designations. Here we present a discussion of the nature of LCDB data, i.e., which values are actually measured and which are derived. For derived data, we give our justification for specific values. We also present some analysis based on the LCDB data, including new default albedo (pV) and phase slope parameter (G) values for the primary taxonomic classes and a review of the frequency-diameter distribution of all asteroids as well as some selected subsets. The most recent version of data used in this analysis is available for download from the Collaborative Asteroid Lightcurve Link (CALL) site at http://www.MinorPlanetObserver.com/astlc/default.htm. Other data sets, some only subsets of the full LCDB, are available in the Ephemeris of Minor Planets, The Planetary Data System, and the Minor Planet Center web site.     

Photometry of Aten asteroids—More than a handful of binaries

Photometric observations were conducted on eight Aten near-Earth asteroids, with the goal of building physical models for the objects (85989) 1999 JD₆, (86450) 2000 CK₃₃, (86667) 2000 FO₁₀, (137170) 1999 HF₁, 1999 MN, 2000 PJ₅, 2002 JC and 2003 NZ₆. The results show rotation periods from 2.3 to almost 26 h. Some objects exhibit amplitudes higher than one magnitude on their lightcurves (1999 JD₆, 2000 CK₃₃ and 2003 NZ₆). Phase curve values (β, H^′, see below) were derived for four Atens and H-G values were found for two. Five Atens were classified by using their B-V and V-R color indices. This taxonomy was compared with the phase coefficient-albedo correlation defined by Belskaya and Shevchenko [Belskaya, I.N., Shevchenko, V.G., 2000. Icarus 147, 94-105]. Color variations during rotation and phase angle change were searched for. Our study demonstrates the high variety among Atens. Five out of the eight Atens are binaries or possible binaries, a significantly higher fraction than the expected 15% [Bottke, W.F., Melosh, H.J., 1996. Nature 281, 51-53]. The lightcurve of 2000 PJ₅ exhibits a binary character with a probable highly eccentric orbital rotation of the secondary component. The different periods of the known binary 1999 HF₁'s are easily detected. Other Atens have lightcurve with features such as high amplitude, V-shaped minima and U-shaped maxima that can be interpreted as a binary asteroid with a synchronous rotation (2003 NZ₆, 2000 CK₃₃ and perhaps also 1999 JD₆). The very red colors of 2000 CK₃₃ suggest a unique surface composition for this near-Earth object.     

Convex profiles from asteroid lightcurves

Integral geometry is used to solve a two-dimensional simplification of the three-dimensional lightcurve inversion problem, and a method is introduced for obtaining a convex profile, $\text{P}$, from asteroid lightcurve data. Whenever four ideal conditions are satisfied, $\text{P}$ is an estimator for the asteroid's “mean cross section,” $\text{C}$, a convex set defined as the average of all cross sections $\text{C}$ cut by planes a distance z above the asteroid's equatorial plane. $\text{C}$ is therefore a two-dimensional average of the asteroid's three-dimensional shape. The ideal conditions are that (A) each curve $\text{C}\text{(z)}$ is convex. (B) the asteroid's scattering law is uniform and geometric, (C) the astrocentric declinations of the Sun and Earth are zero, and (D) the solar phase angle θ ≠ 0. If Condition C is known to hold, the extend to which the lightcurve can be accounted for by a geometrically scattering convex object can be quantified in terms of an appropriate “goodness-of-fit” static. If the solar phase angle is zero, as for radar “lightcurve,” then (i) method yields a profile $\text{P}{}_{\mathrm{<mtext>s</mtext>}}$ the symmetrization $\text{C}{}_{\mathrm{<mtext>s</mtext>}}$ of $\text{C}$; (ii) Condition A need not hold and if it does not, then the inversion yields the symmetrization of the asteroid's mean convex hull; and (iii) Fourier analysis of the lightcurve can reveal violation of Condition B. Doppler-frequency resolution of radar echoes at several rotational phases adds information by constraining the convex hull $\text{H}{}_{\mathrm{<mtext>p</mtext>}}$ of the asteroid's (not necessarily convex) polar silhouette. Estimation of a convex profile from a photoelectric or radar lightcurve is a problem in weighted-least-squares optimization subject to inequality constraints. The solution uses a recursive quadratic programming algorithm to derive a Fourier parameterization for $\text{P}$ from the coefficients in the lightcurve's Fourier expansion. The method has been tested by inverting analytically generated lightcurves for geometrically scattering ellipsoids with semiaxes a ? b ? c, and the inversion yields $\text{P}\text{=}\text{P}{}_{\mathrm{<mtext>s</mtext>}}\text{?}\text{C}\text{=}\text{C}{}_{\mathrm{<mtext>s</mtext>}}\text{=}\text{H}{}_{\mathrm{<mtext>p</mtext>}}$ when the viewing geometry (Condition C) is close to ideal. For situations when the asteroid's pole direction is unknown, a test is offered of the hyphothesis that a given lightcurve can be due to a geometrically scattering ellipsoid with $\text{a}\text{c}\text{? ?}$, where ? is an priori upper bound on the maximum axis ratio. Convex profiles are presented for 15 Eunomia, 118 Peitho, 246 Asporina, 281 Lucretia. 790 Pretoria, 1685 Toro, and 1978 CA.     

Revised albedos of Trojan asteroids (911) Agamemnon and (4709) Ennomos

CCD‐photometry was performed for two Jupiter Trojan asteroids (911) Agamemnon and (4709) Ennomos for which the diameters were obtained from occultation events. New data on rotation periods, lightcurve amplitudes, color indices, magnitude–phase slopes, and absolute magnitudes were obtained for these asteroids. We have used the diameters from occultations (166 and 99 km) and new data on absolute magnitudes at the instant occultation (7.95 and 8.85 mag) to revise their albedos to 0.042 (911 Agamemnon) and 0.052 (4709 Ennomos).     

Angular momentum drain: A mechanism for despinning asteroids

It is proposed that a new mechanism—angular momentum drain—helps account for the relatively slow rotation rates of intermediate-sized asteroids. Impact ejecta on a spinning body preferentially escape in the direction of rotation. This material systematically drains away spin angular momentum, leading to the counterintuitive result that collisions can reduce the spin of midsized objects. For an asteroid of mass M spinning at frequency ω, a mass loss δM correspond to an average decrease in rotation rate $\text{δω ≈}\text{ωδM}\text{M}$. A. W. Harris' (1979), Icarus40, 145–153) theory for the collisional evolution of asteroidal spins is significantly altered by inlusion of this effect. While the modified theory is still somewhat artificial, comparison of its predictions with the data of S. F. Dermott, A. W. Harris, and C. D. Murray (1984, Icarus57, 14–34) suggests that angular momentum drain is essential for understanding the statistics of asteroidal rotations.     

Asteroid rotation: III. 1978 Observations

Photoelectric observations of 32 asteroids observed from Table Mountain Observatory during the second half of 1978 are reported. Rotation periods were obtained for most objects. Absolute magnitudes and phase functions were not determined for any of these asteroids. The geometric mean rotation period of the 32 asteroids observed is 14.2 ± 1.6 hr, as compared to 9.38 ± 0.35 hr for 182 asteroids analyzed in Paper I (A. W. Harris and J. A. Burns, 1979, Icarus 40, 115–144). We attribute this difference to an observational selection effect which favors detection of fast rotation, as discussed in Paper I. If this is true, then the present sample contains the reverse bias, since it is complete in that a period (in some cases very approximate) was obtained for each object observed, but fast rotators are underrepresented due to prior discovery of their rotation properties.     

The rotation of small asteroids

The addition of the unbiased sample of R. P. Binzel and J. D. Mulholland (Icarus56, 519–533) nearly triples the sample size of photoelectrically determined rotational parameters for main belt asteroids with estimated diameters (D) ≤30 km. Nonparametric stattistical tests which require no assumptions about the distributions or variances of the samples are used to examine rotational parameters for all D ≤ 30 km asteroids. A comparison of photoelectric and photographic results shows that the techniques have a highly significant difference in the range of detected frequencies. This difference does not allow photographic and photoelectric observations to be combined for meaningful statistical tests since a photographic bias toward smaller sample variances can induce statistical results that appear overly significant. Photographic observations also show a highly significant bias toward detecting asteroids with larger lightcurve amplitudes. The fit of a Maxwellian to the observed rotational frequency distribution can be rejected at a highly significant confidence level but the observed distribution can be acceptably fit by two Maxwellian distributions, which is consistent with the hypothesis that there are separate populations of slow and fast rotating asteroids. The frequency distributions of <15 km main belt asteroids and Earth and Mars crossers are not found to differ significantly. However, the larger mean lightcurve amplitude of the Earth and Mars crossing asteroids is found to be statistically significant. This latter result is interesting in view of the lack of any strong inverse amplitude versus diameter relation for small asteroids. No significant diameter dependence on rotational frequency is seen among only D ≤ 30 km asteroids. However, the inverse frequency versus diameter relation for D ≤ 120 km asteroids found by S. F. Dermott, A. W. Harris, and C. D. Murray (Icarus, in press) is found to be statistically significant using a linear least-squares analysis of photoelectric data only. No significant diameter dependence on rotational lightcurve amplitude is seen using linear least-squares analysis of photoelectric data for D≤30 and D≤90 km asteroids. However, a significant inverse amplitude versus diameter relation is found when this analysis is extended to D≤120 km asteroids. This finding may be consistent with the hypothesis of Dermott et al. that near 120 km there is a transition between primordial asteroids and their collisional fragments.     

Micropulsations and orthogonal curvilinear coordinates

The theory of torsional hydromagnetic oscillations of the magnetosphere is usually cast in terms of orthogonal curvilinear coordinates. For a general magnetic field B with potential Ω it is shown that no coordinates exist in which a suitable solution may be found unless the Alfvén velocity V_A, together with B and Ω satisfy certain functional relationships. In the case V_A = constant, for example we must have

$\text{(}\text{B}\text{· ?)B =}\text{function of}\text{B}\text{and}\text{Ω}\text{only}$

. The relationships presented are in fact satisfied by all the magnetic fields considered to date.     

Photometry of Phoebe

Observatios of Phoebe (S9) in the V filter at small solar phase angles (0.2° to 1.2°) with the MIT SNAPSHOT CCD are presented. The value of Phoebe's sideral rotational period is refined to 9.282 ± 0.015hr. Assuming the Voyager-derived 110 km radius, Phoebe's observed mean opposition V magnitude of 16.176 ± 0.033 (extrapolated from small angles) corresponds to a geometric albedo of 0.084 ± 0.003. A strong opposition effect is indicated by the 0.180 ± 0.035 mag/deg solar phase coefficient observed at these small phase angles. The data are shown to be compatible with a phase function for C-type asteroids (K. Lumme and E. Bowell, 1981, Astron. J. 86, 1705–1721; K. Lumme, E. Bowell, and A. W. Harris, 1984, Bull. Amer. Astron. Soc. 16, 684), but give a poorer fit to the average asteroid phase relation of T. Gehrels and E.F. Tedesco (1979, Astron. J. 84, 1079–1087). Phoebe's rotational lightcurve in the V filter is roughly sinusoidal, with a 0.230-mag peak-to-peak amplitude and weaker higher order harmonics indicating primarily bimodal surface feature contrast. In addition to these photometric results, precise positions on 3 nights are given.     

The asteroids as outcomes of catastrophic collisions

The role of catastrophic collisions in the evolution of the asteroids is discussed in detail, employing extrapolations of experimental results on the outcomrs of high-velocity impacts. We determine the range of the probable largest collision for target asteroids of different sizes during the solar system's lifetime, and we conclude that all the asteroids have undergone collisional events capable of overcoming the material's solid-state cohesion. Such events do not lead inescapably to complete disruption of the targets, because (i) for a previously unfractured target, experiments show that fragments of significant size can survive breakup, depending on the energy and geometry of the collision; (ii) self-gravitation can easily cause a reaccumulation of fragments for targets exceeding a critical size, which seems to be of the order of 100 km. In the intermediate diameter range 100?D ?300 km, where formation of gravitationally bound “rubble piles” is frequent, the transfer of angular momentum can be large enough to produce objects with triaxial equilibrium shapes (Jacobi ellipsoids) or to cause fission into binary systems. In the same size range, low-velocity escape of collisional fragments can also occur, leading to the formation of dynamical families. Asteroids smaller than ～100 km are mostly multigeneration fragments, while for $\text{D?300}\text{km}$ the collisional process produces nearly spheroidal objects covered by megaregoliths; whether their rotation is “primordial” or collisionally generated depends critically on the past flux of colliders. The complex and size-dependent phenomenology predicted by the theory compares satisfactorily with the observational evidence, as derived both by a classification of asteroids in terms of their size, spin rate, and lightcurve amplitude, and by a comparison between the rotational properties of family and nonfamily asteroids. The fundamental result of this investigation is that almost all asteroids are outcomes of catastrophic collisions, and that these events cause either complete fragmentation of the target bodies or, at least, drastic readjustments of their internal structure, shape, and spin rate.     

The Diagnosis of Complex Rotation in the Lightcurve of 4179 Toutatis and Potential Applications to Other Asteroids and Bare Cometary Nuclei

The analysis of radar observations of the asteroid 4179 Toutatis by Hudson and Ostro (1995, Science270, 84-86) yielded a complex spin state. We revisit the visible lightcurve data on Toutatis (Spencer et al. 1995, Icarus117, 71-89) to explore the feasibility of using a rotational lightcurve to recover the signature of an excited spin state. For this, we apply Fourier transform and CLEAN algorithm (WindowCLEAN). WindowCLEAN yields clear and precise frequency signatures associated with the precession of the long axis about the total angular momentum vector and a combination of this precession and rotation about the long axis. For a long-axis mode state, our periodicities for Toutatis yield a mean long-axis precession period, P_φ, of 7.38 days and a rotation period around the long axis, P_ψ, of 5.38 days, which compare well with the respective periods of 7.42 and 5.37 days derived by Ostro et al. (1999, Icarus137, 122-139) and represent an independent confirmation of these values. We explain why the dramatic change in the Earth-Toutatis-Sun geometry during the time that the lightcurve was obtained has little effect on the final results obtained. Using the Toutatis example as a guide, we discuss the capabilities as well as the limitations on deriving information about complex spin states from asteroidal lightcurves.     

Photoelectric lightcurves of the asteroid 1862 Apollo

Photoelectric lightcurves of the asteroid 1862 Apollo were obtained in November–December 1980 and in April–May 1982. The period of rotation is unambiguously determined to be 3.0655 ± 0.0008 hr. The 1980 observations span a range of solar phase angle from 30° to 90°, and the 1982 observations, 0.°2 to 90°. The Lumme-Bowell-Harris phase relation can be fit to the absolute magnitudes at maximum light with an RMS scatter of 0.06 magnitude over the entire range of phase angle. The constants of the solution are absolute V magnitude at zero phase angle and at maximum light, 16.23 ± 0.02; slope parameter, 0.23 ± 0.01. These constant corresponds to values in the linear phase coefficient system of V(1, 0) = 16.50 ± 0.02 and a phase coefficient of β_v = 0.0305 ± 0.0012 mag/degree in the phase range 10°–20°. The slope of the phase curve is typical for a moderate albedo asteroid. The absolute magnitudes observed in 1980 and 1982 fall along a common phase curve. That is, Apollo was not intrinsically brighter at one apparition than the other. This is not surprising, since the two apparitions were almost exactly opposite one another in the sky. A pole position was calculated from the observed deviation of the lightcurve from constant periodicity (synodic-sidereal difference) during both apparitions. The computed 1950 ecliptic coordinates of the pole are: longitude = 56°, latitude = −26°. This is the “north” pole with respect to right-handed (counter-clockwise) rotation. The formal uncertainty of the solution for the pole position is less than 10°, but realistically may be several times that, or even completely wrong. The sidereal period of rotation asscociated with this pole solution is 3.065436 ± 0.000012 hr.     

Chaotic behavior and the origin of the 31 Kirkwood gap

The sudden eccentricity increases discovered by J. Wisdom (Astron J.87, 577–593, 1982) are reproduced in numerical integrations of the planar-elliptic restricted three-body problem, verifying that this phenomenon is real. Maximum Lyapunov characteristic exponents for trajectories near the $\text{3}\text{1}$ commensurability are computed both with the mappings presented in Wisdom (1982) and by numerical integration of the planar-elliptic problem. In all cases the agreement is excellent, indicating that the mappings accurately reflect whether trajectories are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the $\text{3}\text{1}$ commensurability, both in the planar-elliptic problem and to a more limited extent in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the $\text{3}\text{1}$ Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.     

1580 Betulia: An unusual asteroid with an extraordinary lightcurve

Results of UBV photometry and polarimetry of 1580 Betulia during its 1976 apparition are presented. The synodic period of rotation is found to be 6.130 hr. The linear phase coefficient and absolute magnitude of the primary maximum in V are 0.032 mag/deg and 14.88, respectively. No color variations with rotation or solar phase angle detected, the mean colors being B?V = 0.66 and U?B = 0.24. Betulia's lightcurve is unique among asteroids studied to date in that it displays three maxima and three minima within one rotational cycle, indicative of a region of greater roughness and/or a dark spot on one of its broad faces. Polarization results indicate a low albedo and a mean diameter of about 7 km, establishing Betulia as the first C type asteroid to be found among the Mars crossers. A model accounting for most features of Betulia's lightcurve is given by a prolate spheroid rotating about one of its shorter axes having an axis ratio of 1:1.21 with a major topographic feature on one of its broad faces.     

UBV photometry of Vesta

A lightcurve of Vesta, obtained on four nights between June 23 and 30 June 1978 during the coordinated campaign for the determination of the rotation period, is presented. The observations were performed at the 91-cm telescope of the Catania Observatory employing UBV filters and a photon counting photometer. The V lightcurve apparently shows a single maximum, suggesting that the 5^h20^m29^s.2 period is the correct one. Features are evident near the maximum and the minimum closely resembling those of Johnson's lightcurve of 22 December 1950 and Taylor's of January 21, 1973. The amplitude in V light is 0^m.105 and small variations are also found in the color indices. The largest color variation is for the U-V with Δm = 0.^m.05, which is slightly larger than the value 0^m.02 found by T. Gehrels [Astron. J.72, 929 (1967)]. The maximum and minimum values occur at the same phase with respect to the maximum V light as found by Gehrels, i.e., Vesta appears bluer near 0^p.25 and redder near 0^p.7. Corrections with the solar phase angle were made using the coefficients given by Gehrels for the B-V and U-V while a new value of 0.036 mag/deg was assumed for the V observations. The available amplitudes of Vesta's lightcurve were analyzed with respect to the longitude position and the solar phase angle.     

Asteroids with large secular orbital variations

As the classical linear theory of secular perturbations for asteroids is known not to be adequate for computing the perturbations of asteroids with high eccentricities and/or inclinations, a seminumerical method to calculate the secular perturbations by including higher-degree terms in the disturbing function has been developed. It is here applied to asteroids with small values of $\text{(1 ? e}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{cos}\text{i}$, since the secular variations as well as their deviations from the results derived by the classical linear theory are generally large for such asteroids. It is found that the arguments of perihelion for five of the numbered asteroids are librating around 90 or 270°. For asteroids with $\text{(1 ? e}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>cos</mtext><mtext>\; i</mtext>}}$ less than 0.85 the results of the secular variations are tabulated. Also the stability of such orbits is discussed by comparing the orbital properties of short-periodic comets with them. Generally speaking, orbits of the asteroids are more stable than those of the short-periodic comets, and asteroids with librating arguments of perihelion are more stable than those with circular coplanar orbits although their orbital elements are changed more by secular perturbations.     

Non-adiabatic processes in a two-fluid plasma and energization of the plasma sheet plasma

The change of energy of a collisionless, two-fluid plasma consists of the adiabatic gain or loss of energy, which is due to the work done by the electromagnetic forces, and of the non-adiabatic change associated with the presence of the “rest” field $\text{E}{}^1\text{=}\text{E}\text{+ (}\text{1}\text{c}\text{)}\text{V}\text{×}\text{B}$. The non-adiabatic gain or loss of energy per unit ti may be expressed by the relation

$\text{Q=}\text{E}{}_{\mathrm{\parallel }}\text{·}\text{i}{}_{\mathrm{\parallel }}\text{+}\text{c}\text{eNB}{}^2\text{B·}\text{f}\text{?}\text{×f}$

where i is the density of conductive current, N the ion number-density, and f (f?) the sum of inertia and pressure divergence of ions (electrons). Symbols of parallelism refer to the direction of B.A special case of non-adiabatic energization of a slowly convecting plasma sheet plasma is discussed in some detail. Regardless of the value of V, the non-adiabatic energization may significantly exceed any conceivable energization associated with the electric field $\text{?(}\text{1}\text{c}\text{)}\text{V}\text{×}\text{B}$.