- Email: [email protected]
The Shape of Eros from NEAR Imaging Data
Icarus 145, 348–350 (2000) doi:10.1006/icar.2000.6406, available online at http://www.idealibrary.com on
The Shape of Eros from NEAR Imaging Data P. C. Thomas, J. Joseph, B. Carcich, B. E. Clark, and J. Veverka CRSR, Cornell University, Ithaca, New York 14853 E-mail: [email protected]
J. K. Miller, W. Owen, and B. Williams Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109
and M. Robinson Northwestern University, Department of Geological Sciences, 1847 Sheridan Road, Evanston, Illinois 60208-2150 Received October 13, 1999; revised January 4, 2000
3. SHAPE RESULTS Images obtained during the NEAR spacecraft flyby of 433 Eros in December 1998 reveal a curved object, of a mean size and shape close to those inferred from lightcurve and radar data. The resolution of the images was sufficient to identify landmarks and establish a surface-based coordinate system. °c 2000 Academic Press
1. INTRODUCTION
The unscheduled flyby of Asteroid 433 Eros on December 23, 1998 by the NEAR spacecraft provided an opportunity to obtain imaging, spectral, and radio science data on this object (Veverka et al. 1999; Yeomans et al. 1999) and to develop a preliminary shape model. In this note we report on the characteristics of the shape and the prime meridian definition and briefly discuss some of the scientific and operational implications of the shape. 2. DATA AND TECHNIQUES
The flyby occurred with the sun at about −32◦ latitude on Eros. Observations began with the spacecraft at 52◦ N, at a phase angle of about 85◦ . The closest approach was at 3827 km, with a mean pixel scale of about 470 m. (The NEAR imager is described by Veverka et al. (1998) and Murchie et al. (1999).) Outbound imaging is at phase angles generally over 110◦ . The range of spacecraft latitudes and Eros’ rotation allowed for imaging of more than 2/3 of the asteroid’s surface. As shown in Fig. 1, positions of surface features provided data for a solution of the spin pole orientation and control point locations. The solution was obtained in the manner described by Davies et al. (1996) and Thomas et al. (1994). Yeomans et al. (1999) included a control point solution in a more general navigation calculation that included solutions for Eros-relative positions and Eros’ mass.
The spin pole solution (Table I) of RA = 11.9◦ , Dec = 20.8◦ (±4◦ ) (J2000) was derived from 19 surface features identified in 21 images (92 measurements). This solution is within 5◦ of that inferred previously from ground-based lightcurve studies summarized by Zellner 1976, and within 5◦ of the independent solution of Yeomans et al. (1999). Limb and terminator positions provide most of the shape data; the greatest uncertainties in the derived shape and size arise from the lack of illumination at high northern latitudes and from low resolution control of southern limbs relative to the northern ones caused by a gap in image coverage as the spacecraft rapidly changed latitude. Figure 1 shows the position of the adopted 0◦ longitude line. Table II summarizes the relation of this longitude convention to previous ones, similar to the summary table in Mitchell et al. (1996). The longitude convention adopted is very close to that of Murchie and Pieters (1996). The volume found, in combination with the mass reported by Yeomans et al. (1999), yields a mean density of 2.5 ± 0.7 g cm−3 . The shape is sufficiently nonellipsoidal so that fitting an ellipsoid gives misleading mean parameters. All the parameters in Table I are based on the 5◦ × 5◦ shape model. The longest dimension through the model is 31 km. Moments of inertia calculated assuming a uniform distribution of mass yield a maximum moment orientation (principal axis) within 5◦ of the calculated spin pole (z axis) and a minimum moment within 8◦ longitude of the model x axis (0◦ , 0◦ ). Figure 2 and the difference between the principal axes and the model z axis suggest the level of likely error in shape/spin pole determinations at this time, which amounts to about 4◦ and local radius errors of ∼1 km in some areas. As noted by Veverka et al. (1999), the outline obtained from the imaging data is similar to the profile derived by Mitchell et al.
348 0019-1035/00 $35.00 c 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
349
THE SHAPE OF EROS
TABLE II Rotational and Longitude Conventions M & P (1996) and this work phase
Millis et al. (1976) phase
Ostro et al. (1990) phase
0◦ 90◦ 180◦ 270◦
Min 2 Max 1 Min 1 Max 2
180◦ 270◦ 0◦ 90◦
Note. M & P: Murchie and Pieters (1996).
by Murchie and Pieters (1996); the putative olivine-rich side is the convex side, roughly centered on 270◦ . Clark et al. (2000) report lightcurve studies from NEAR images before and after the flyby that provide additional constraints on the epoch and spin rate definition as reported here. 4. IMPLICATIONS OF THE SHAPE
Of the four asteroids imaged close-up by spacecraft, two have distinctly bent or banana shapes (Ida and Eros). Some radar-imaged objects show suggestions of being bilobed or bent (Hudson and Ostro 1994, 1995). While the bent shapes suggest assembled discrete fragments, in detail the images of both Ida and Eros are not easily reconciled with simple two-component objects. Galileo obtained very high resolution images of the area FIG. 1. View of Eros and longitude convention. Visible length is about 30 km. Image 89846123 with shape model displayed at 15◦ intervals in both latitude and longitude. The 0◦ longitude is designated; 270◦ W is at the right side of the visible disk.
(1996) from radar data, in being somewhat banana-shaped. The profiles are shown in part of Fig. 2. The longitude definition follows IAU conventions described in Davies et al. (1994). The prime meridian is defined by W = 324.08◦ + 1639.3922d, where W is the distance of the prime meridian eastward from the intersection of the object’s equator and the standard earth equator, and d is the time in days from the standard epoch (2000 January 1.5; JD 2451545.0, TDB). Figure 2 includes the equatorial profile labeled with the lightcurve designations of Millis et al. (1976). Longitudes roughly centered on 90◦ are the presumably pyroxene-rich side detected TABLE I Eros Shape Model, Spin Pole Results Volume Mean radius Minimum radius from center of figure Maximum radius from center of figure Spin pole
2900 km3 ± 600 km3 8.8 km ± 0.6 km 3.9 km 16.4 km RA = 11.9◦ , Dec = 20.8◦ ± 4◦ J2000 W = 324.08◦ + 1639.3922d
Note. W defines the prime meridian; see text and Davies et al. (1994).
FIG. 2. Shape of Eros. Left side: Views of the shape model grid. The shape model is viewed from the principal directions defined by the spin pole and the 0◦ longitude (slightly different from principal moment axes; see text). Areas above about 40◦ N were not illuminated, so the shape model in this area is constrained only by lack of visibility. The view from the north pole includes the lightcurve convention of Millis et al. (1976). m1 and m2 are minima; M1 and M2 are maxima; see Table II for other published conventions. Right side: Physical and gravitational shapes of Eros. Top is the physical equatorial profile of Eros. Middle is the gravitational shape if Eros did not rotate, calculated by the method of Thomas (1993) to give relative topography, which has then been added to a sphere of Eros’ mean radius (8.8 km). Bottom shows the gravitational shape with the rotational accelerations included. See text for background.
350
THOMAS ET AL.
of Ida’s “bend,” and this heavily cratered area showed no indications expected of separate pieces, or regolith-covered pieces (Sullivan et al. 1996). For Eros, the presence of a continuous feature, a ridge or step, running the length of the object also seems at odds with Eros consisting of accumulated large fragments (Veverka et al. 1999). In the case of Eros, there is hope of testing this interpretation with 100-fold improvement in imaging resolution in 2000. Eros’ shape is useful in illustrating the effects that contribute to relative gravitational positions on irregularly shaped objects. On rapidly rotating, irregularly shaped objects, what is gravitationally up or down may not be intuitively obvious from just observing the physical shape of the object. Quantitative determination of relative heights needs calculation of acceleration at the surface specifically accounting for rotation and the nonpoint source mass distribution (Thomas 1993). One can calculate the angle between the local surface normal and surface acceleration to obtain the slope (for instance, Thomas et al. 1996). Maps of slopes, however, do not easily convey a global picture of the gravitational shape. An alternative method (Thomas 1993) is to map the relative potential energy at the surface, or the geopotential number, and thus get relative gravitational heights (strictly, the dynamic height obtained by dividing differences in potential energy by a constant reference gravity; see Vanicek and Krakiwsky 1986). Figure 2 includes the result of such a calculation for Eros (mean density 2.5 g cm−3 , Yeomans et al. 1999) in two steps for a generalized illustrative purpose. First, no rotation was assumed, and only the effects of the shape and assumed uniform mass distribution were made. The middle equatorial profile of Fig. 2 shows the result if the relative gravitational heights along the equator are placed on an sphere of the same mean radius (8.8 km) as Eros. This result emphasizes that the mass distribution alone makes the effective topography much more subdued than the changing radii might suggest. The bottom profile of Fig. 2 shows the result if rotational accelerations (spin period of Eros is 5.27 h) are added. The topography becomes even more subdued because of the effects of the greater rotational accelerations at the long ends of the object. The shape of Eros presents challenges for close spacecraft operations, as the gravity field at distances desired for lengthy operations is likely to have significant high-order components. The elongated shape may, however, provide for close flybys of the ends for very high-resolution imaging. ACKNOWLEDGMENTS The Mission Design, Mission Operations, and Spacecraft teams of the NEAR project at the Applied Physics Laboratory of Johns Hopkins University made possible the return of data from this flyby.
REFERENCES Clark, B. E., P. C. Thomas, J. Veverka, P. Helfenstein, M. S. Robinson, S. L. Murchie 2000. NEAR lightcurves of Asteroid 433 Eros. Icarus. 145, 641– 644. Davies, M. E., V. K. Abalakin, M. Bursa, J. H. Lieske, B. Morando, D. Morrison, P. K. Seidelmann, A. T. Sinclair, B. Yallop, and Y. S. Tjuflin 1994. Report of the IAU/IAG/CORPAR working group on cartographic coordinates and rotational elements of the planets and satellites: 1994. Celest. Mech. Dynam. Astron. 63, 127–148. Davies, M. E., T. R. Colvin, M. J. S. Belton, J. Veverka, and P. C. Thomas 1996. The direction of the north pole and control network of Asteroid 243 Ida. Icarus 120, 33–37. Hudson, R. S., and S. J. Ostro 1994. Shape of Asteroid 4769 Castalia (1989 PB) from inversion of radar images. Science 263, 940–943. Hudson, R. S., and S. J. Ostro 1995. Shape and non-principal axis spin state of Asteroid 4179 Toutatis. Science 270, 84–86. Millis, R. L., E. Bowell, and D. T. Thompson 1976. UBV photometry of Asteroid 433 Eros. Icarus 28, 53–67. Mitchell, D. L., R. S. Hudson, S. J. Ostro, and K. D. Rosema 1998. Shape of Asteroid 433 Eros from inversion of Goldstone radar Doppler spectra. Icarus 131, 4–14. Murchie, S. L. and C. M. Pieters 1996. Spectral properties and rotational spectral heterogeneity of 433 Eros. J. Geophys. Res. 101, 2201–2214. Murchie, S., M. S. Robinson, E. Hawkins, K. Peacock, B. E. Clark, P. C. Thomas, J. F. Bell III, and J. Veverka 1999. Inflight calibration and performance of the NEAR multispectral imager. Icarus 140, 66– 91. Ostro, S. J., K. D. Rosema, and R. F. Jurgens 1990. The shape of Eros. Icarus 84, 334–351. Simonelli, D. P., P. C. Thomas, B. T. Carcich, and J. Veverka 1993. The generation and use of numerical shape models for irregular Solar System objects. Icarus 103, 49–61. Sullivan, R., and 17 colleagues 1996. Geology of 243 Ida. Icarus 120, 106– 118. Thomas, P. 1993. Gravity, tides, and topography on small satellites and asteroids. Icarus 105, 326–344. Thomas, P. C., J. Veverka, D. Simonelli, P. Helfenstein, B. Carcich, M. J. S. Belton, M. E. Davies, and C. Chapman 1994. The shape of Gaspra. Icarus 107, 23–36. Vanicek, P., and E. J. Krakiwsky 1986. Geodesy: The Concepts, 2nd ed. Elsevier, New York. Veverka, J., J. F. Bell III, P. Thomas, A. Harch, S. Murchie, S. E. Hawkins III, J. Warren, H. Darlington, K. Peacock, C. R. Chapman, L. A. McFadden, M. C. Malin, and M. S. Robinson 1997. An overview of the NEAR multispectral imager-near-infrared spectrometer investigation. J. Geophys. Res. 102, 23,709–23,727. Veverka, J. and 25 colleagues 1999. Imaging of Asteroid 433 Eros during NEAR’s flyby reconnaissance. Science 285, 562–564. Yeomans, D. K., and 14 colleagues 1999. Estimating the mass of Asteroid 433 Eros during the NEAR spacecraft flyby. Science 285, 560– 562. Zellner, B. 1976. Physical properties of Asteroid 433 Eros. Icarus 28, 149– 153.
The Shape of Eros from NEAR Imaging Data P. C. Thomas, J. Joseph, B. Carcich, B. E. Clark, and J. Veverka CRSR, Cornell University, Ithaca, New York 14853 E-mail: [email protected]
J. K. Miller, W. Owen, and B. Williams Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109
and M. Robinson Northwestern University, Department of Geological Sciences, 1847 Sheridan Road, Evanston, Illinois 60208-2150 Received October 13, 1999; revised January 4, 2000
3. SHAPE RESULTS Images obtained during the NEAR spacecraft flyby of 433 Eros in December 1998 reveal a curved object, of a mean size and shape close to those inferred from lightcurve and radar data. The resolution of the images was sufficient to identify landmarks and establish a surface-based coordinate system. °c 2000 Academic Press
1. INTRODUCTION
The unscheduled flyby of Asteroid 433 Eros on December 23, 1998 by the NEAR spacecraft provided an opportunity to obtain imaging, spectral, and radio science data on this object (Veverka et al. 1999; Yeomans et al. 1999) and to develop a preliminary shape model. In this note we report on the characteristics of the shape and the prime meridian definition and briefly discuss some of the scientific and operational implications of the shape. 2. DATA AND TECHNIQUES
The flyby occurred with the sun at about −32◦ latitude on Eros. Observations began with the spacecraft at 52◦ N, at a phase angle of about 85◦ . The closest approach was at 3827 km, with a mean pixel scale of about 470 m. (The NEAR imager is described by Veverka et al. (1998) and Murchie et al. (1999).) Outbound imaging is at phase angles generally over 110◦ . The range of spacecraft latitudes and Eros’ rotation allowed for imaging of more than 2/3 of the asteroid’s surface. As shown in Fig. 1, positions of surface features provided data for a solution of the spin pole orientation and control point locations. The solution was obtained in the manner described by Davies et al. (1996) and Thomas et al. (1994). Yeomans et al. (1999) included a control point solution in a more general navigation calculation that included solutions for Eros-relative positions and Eros’ mass.
The spin pole solution (Table I) of RA = 11.9◦ , Dec = 20.8◦ (±4◦ ) (J2000) was derived from 19 surface features identified in 21 images (92 measurements). This solution is within 5◦ of that inferred previously from ground-based lightcurve studies summarized by Zellner 1976, and within 5◦ of the independent solution of Yeomans et al. (1999). Limb and terminator positions provide most of the shape data; the greatest uncertainties in the derived shape and size arise from the lack of illumination at high northern latitudes and from low resolution control of southern limbs relative to the northern ones caused by a gap in image coverage as the spacecraft rapidly changed latitude. Figure 1 shows the position of the adopted 0◦ longitude line. Table II summarizes the relation of this longitude convention to previous ones, similar to the summary table in Mitchell et al. (1996). The longitude convention adopted is very close to that of Murchie and Pieters (1996). The volume found, in combination with the mass reported by Yeomans et al. (1999), yields a mean density of 2.5 ± 0.7 g cm−3 . The shape is sufficiently nonellipsoidal so that fitting an ellipsoid gives misleading mean parameters. All the parameters in Table I are based on the 5◦ × 5◦ shape model. The longest dimension through the model is 31 km. Moments of inertia calculated assuming a uniform distribution of mass yield a maximum moment orientation (principal axis) within 5◦ of the calculated spin pole (z axis) and a minimum moment within 8◦ longitude of the model x axis (0◦ , 0◦ ). Figure 2 and the difference between the principal axes and the model z axis suggest the level of likely error in shape/spin pole determinations at this time, which amounts to about 4◦ and local radius errors of ∼1 km in some areas. As noted by Veverka et al. (1999), the outline obtained from the imaging data is similar to the profile derived by Mitchell et al.
348 0019-1035/00 $35.00 c 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
349
THE SHAPE OF EROS
TABLE II Rotational and Longitude Conventions M & P (1996) and this work phase
Millis et al. (1976) phase
Ostro et al. (1990) phase
0◦ 90◦ 180◦ 270◦
Min 2 Max 1 Min 1 Max 2
180◦ 270◦ 0◦ 90◦
Note. M & P: Murchie and Pieters (1996).
by Murchie and Pieters (1996); the putative olivine-rich side is the convex side, roughly centered on 270◦ . Clark et al. (2000) report lightcurve studies from NEAR images before and after the flyby that provide additional constraints on the epoch and spin rate definition as reported here. 4. IMPLICATIONS OF THE SHAPE
Of the four asteroids imaged close-up by spacecraft, two have distinctly bent or banana shapes (Ida and Eros). Some radar-imaged objects show suggestions of being bilobed or bent (Hudson and Ostro 1994, 1995). While the bent shapes suggest assembled discrete fragments, in detail the images of both Ida and Eros are not easily reconciled with simple two-component objects. Galileo obtained very high resolution images of the area FIG. 1. View of Eros and longitude convention. Visible length is about 30 km. Image 89846123 with shape model displayed at 15◦ intervals in both latitude and longitude. The 0◦ longitude is designated; 270◦ W is at the right side of the visible disk.
(1996) from radar data, in being somewhat banana-shaped. The profiles are shown in part of Fig. 2. The longitude definition follows IAU conventions described in Davies et al. (1994). The prime meridian is defined by W = 324.08◦ + 1639.3922d, where W is the distance of the prime meridian eastward from the intersection of the object’s equator and the standard earth equator, and d is the time in days from the standard epoch (2000 January 1.5; JD 2451545.0, TDB). Figure 2 includes the equatorial profile labeled with the lightcurve designations of Millis et al. (1976). Longitudes roughly centered on 90◦ are the presumably pyroxene-rich side detected TABLE I Eros Shape Model, Spin Pole Results Volume Mean radius Minimum radius from center of figure Maximum radius from center of figure Spin pole
2900 km3 ± 600 km3 8.8 km ± 0.6 km 3.9 km 16.4 km RA = 11.9◦ , Dec = 20.8◦ ± 4◦ J2000 W = 324.08◦ + 1639.3922d
Note. W defines the prime meridian; see text and Davies et al. (1994).
FIG. 2. Shape of Eros. Left side: Views of the shape model grid. The shape model is viewed from the principal directions defined by the spin pole and the 0◦ longitude (slightly different from principal moment axes; see text). Areas above about 40◦ N were not illuminated, so the shape model in this area is constrained only by lack of visibility. The view from the north pole includes the lightcurve convention of Millis et al. (1976). m1 and m2 are minima; M1 and M2 are maxima; see Table II for other published conventions. Right side: Physical and gravitational shapes of Eros. Top is the physical equatorial profile of Eros. Middle is the gravitational shape if Eros did not rotate, calculated by the method of Thomas (1993) to give relative topography, which has then been added to a sphere of Eros’ mean radius (8.8 km). Bottom shows the gravitational shape with the rotational accelerations included. See text for background.
350
THOMAS ET AL.
of Ida’s “bend,” and this heavily cratered area showed no indications expected of separate pieces, or regolith-covered pieces (Sullivan et al. 1996). For Eros, the presence of a continuous feature, a ridge or step, running the length of the object also seems at odds with Eros consisting of accumulated large fragments (Veverka et al. 1999). In the case of Eros, there is hope of testing this interpretation with 100-fold improvement in imaging resolution in 2000. Eros’ shape is useful in illustrating the effects that contribute to relative gravitational positions on irregularly shaped objects. On rapidly rotating, irregularly shaped objects, what is gravitationally up or down may not be intuitively obvious from just observing the physical shape of the object. Quantitative determination of relative heights needs calculation of acceleration at the surface specifically accounting for rotation and the nonpoint source mass distribution (Thomas 1993). One can calculate the angle between the local surface normal and surface acceleration to obtain the slope (for instance, Thomas et al. 1996). Maps of slopes, however, do not easily convey a global picture of the gravitational shape. An alternative method (Thomas 1993) is to map the relative potential energy at the surface, or the geopotential number, and thus get relative gravitational heights (strictly, the dynamic height obtained by dividing differences in potential energy by a constant reference gravity; see Vanicek and Krakiwsky 1986). Figure 2 includes the result of such a calculation for Eros (mean density 2.5 g cm−3 , Yeomans et al. 1999) in two steps for a generalized illustrative purpose. First, no rotation was assumed, and only the effects of the shape and assumed uniform mass distribution were made. The middle equatorial profile of Fig. 2 shows the result if the relative gravitational heights along the equator are placed on an sphere of the same mean radius (8.8 km) as Eros. This result emphasizes that the mass distribution alone makes the effective topography much more subdued than the changing radii might suggest. The bottom profile of Fig. 2 shows the result if rotational accelerations (spin period of Eros is 5.27 h) are added. The topography becomes even more subdued because of the effects of the greater rotational accelerations at the long ends of the object. The shape of Eros presents challenges for close spacecraft operations, as the gravity field at distances desired for lengthy operations is likely to have significant high-order components. The elongated shape may, however, provide for close flybys of the ends for very high-resolution imaging. ACKNOWLEDGMENTS The Mission Design, Mission Operations, and Spacecraft teams of the NEAR project at the Applied Physics Laboratory of Johns Hopkins University made possible the return of data from this flyby.
REFERENCES Clark, B. E., P. C. Thomas, J. Veverka, P. Helfenstein, M. S. Robinson, S. L. Murchie 2000. NEAR lightcurves of Asteroid 433 Eros. Icarus. 145, 641– 644. Davies, M. E., V. K. Abalakin, M. Bursa, J. H. Lieske, B. Morando, D. Morrison, P. K. Seidelmann, A. T. Sinclair, B. Yallop, and Y. S. Tjuflin 1994. Report of the IAU/IAG/CORPAR working group on cartographic coordinates and rotational elements of the planets and satellites: 1994. Celest. Mech. Dynam. Astron. 63, 127–148. Davies, M. E., T. R. Colvin, M. J. S. Belton, J. Veverka, and P. C. Thomas 1996. The direction of the north pole and control network of Asteroid 243 Ida. Icarus 120, 33–37. Hudson, R. S., and S. J. Ostro 1994. Shape of Asteroid 4769 Castalia (1989 PB) from inversion of radar images. Science 263, 940–943. Hudson, R. S., and S. J. Ostro 1995. Shape and non-principal axis spin state of Asteroid 4179 Toutatis. Science 270, 84–86. Millis, R. L., E. Bowell, and D. T. Thompson 1976. UBV photometry of Asteroid 433 Eros. Icarus 28, 53–67. Mitchell, D. L., R. S. Hudson, S. J. Ostro, and K. D. Rosema 1998. Shape of Asteroid 433 Eros from inversion of Goldstone radar Doppler spectra. Icarus 131, 4–14. Murchie, S. L. and C. M. Pieters 1996. Spectral properties and rotational spectral heterogeneity of 433 Eros. J. Geophys. Res. 101, 2201–2214. Murchie, S., M. S. Robinson, E. Hawkins, K. Peacock, B. E. Clark, P. C. Thomas, J. F. Bell III, and J. Veverka 1999. Inflight calibration and performance of the NEAR multispectral imager. Icarus 140, 66– 91. Ostro, S. J., K. D. Rosema, and R. F. Jurgens 1990. The shape of Eros. Icarus 84, 334–351. Simonelli, D. P., P. C. Thomas, B. T. Carcich, and J. Veverka 1993. The generation and use of numerical shape models for irregular Solar System objects. Icarus 103, 49–61. Sullivan, R., and 17 colleagues 1996. Geology of 243 Ida. Icarus 120, 106– 118. Thomas, P. 1993. Gravity, tides, and topography on small satellites and asteroids. Icarus 105, 326–344. Thomas, P. C., J. Veverka, D. Simonelli, P. Helfenstein, B. Carcich, M. J. S. Belton, M. E. Davies, and C. Chapman 1994. The shape of Gaspra. Icarus 107, 23–36. Vanicek, P., and E. J. Krakiwsky 1986. Geodesy: The Concepts, 2nd ed. Elsevier, New York. Veverka, J., J. F. Bell III, P. Thomas, A. Harch, S. Murchie, S. E. Hawkins III, J. Warren, H. Darlington, K. Peacock, C. R. Chapman, L. A. McFadden, M. C. Malin, and M. S. Robinson 1997. An overview of the NEAR multispectral imager-near-infrared spectrometer investigation. J. Geophys. Res. 102, 23,709–23,727. Veverka, J. and 25 colleagues 1999. Imaging of Asteroid 433 Eros during NEAR’s flyby reconnaissance. Science 285, 562–564. Yeomans, D. K., and 14 colleagues 1999. Estimating the mass of Asteroid 433 Eros during the NEAR spacecraft flyby. Science 285, 560– 562. Zellner, B. 1976. Physical properties of Asteroid 433 Eros. Icarus 28, 149– 153.