Lightcurves obtained for 1917 Cuyo at solar phase angles near 54 degrees have an amplitude delta m = 0.44 mag. However, convex-profile inversion of the lightcurves yields an estimate of the asteroid's mean cross section (C, a 2-D average of the 3-D shape) that is only slightly noncircular, with an elongation approximately 1.15. The estimate of C undoubtedly contains systematic errors, the most severe of which could arise from non-equatorial viewing/illumination geometry. However, Cuyo's radar echo shows very little variation in bandwidth vs. rotation phase, supporting the hypothesis that this asteroid's elongation is rather modest

Ostro, Steven J.

Wisniewski, Wielaw Z.

English

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Asteroids, Comets, Meteors 1991, pp. 447450
Lunar and Planetary Institute, Houston, 1992 447
N9
THE SHAPE OF ASTEROID 1917 CUYO
S. J. Ostro
Jet Propulsion Laboratory, California Institute of Technology
and /_ /
W. Z. Wisnlewski
Lunar and Planetary Laboratory, University of Arizona
ABSTRACT
Lightcurves obtained for 1917 Cuyo at solar phase angles near 54 ° have an
amplitude Am = 0.44 mag. However, convex-profile inversion of the
lightcurves yields an estimate of the asteroid's mean cross section (Q,
a 2-D average of the 3-D shape) that is only slightly noncircular, with an
elongation ~ 1.15. The estimate of Q undoubtedly contains systematic errors,
the most severe of which could arise from non-equatorial viewing/illumination
geometry. However, Cuyo's radar echo shows very little variation in bandwidth
vs. rotation phase, supporting the hypothesis that this asteroid's elongation
is rather modest.
INTRODUCTION
The Amor asteroid 1917 Cuyo (1968 AA) passed within 0.15 AU from Earth
in October 1989, during the most favorable apparition for groundbased
astronomy until 2030 (Harvey, 1989). The object was observed by one of
us (WZW) with a CCD camera on the 2.3-meter Steward Observatory telescope on
Kitt Peak and with radar by Ostro et al. (1991) from Arecibo and Goldstone
during a two-week period near closest approach (Table i). Figure 1 shows the
lightcurves and results of inverting those data, and Fig. 2 shows radar echo
spectra. All analyses discussed here use the apparent rotation period,
P = 2.693 h, determined from the lightcurves.
Table i. Observations of Cuyo
RA
Observation UTC dates spanned (h)
Distance (AU)
Dec. Earth Sun
radar (Arecibo)
CCD photometry
radar (Goldstone)
1989 Sap 26 to 30 20.3 9° 0.17
1989 Oct 7 21.0 -13 ° 0.14 1.08
to 8 21.1 -16 ° 0.14 1.08
1989 Oct 9 to ii 21.3 -22 ° 0.14
SHAPE CONSTRAINTS FROM LIGHTCURVES
Under certain ideal conditions, one can use "convex-profile inversion"
of a lightcurve to estimate a profile that is a 2-D _ of the asteroid's
shape (Ostro et al. 1988). That profile is called the mean cross section, _,
and is defined as the average of the envelopes on all the surface contours
parallel to the equator. The ideal conditions for estimating Q include
Condition GEO, that the scattering is uniform and geometric; Condition EVIG,
that the viewing-illumination geometry is equatorial; and Condition PHASE_
that the solar phase angle # is known and nonzero. The logic behind these
conditions is that they collapse the 3-D lightcurve inversion problem, which
cannot be solved uniquely, into a 2-D problem that can.
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448 Asteroids, Comets, Meteors 1991
Condition EVIG, that the sub-Earth and sub-Sun points lie on the
equator, and Condition PHASE imply that _ equals its equatorial component _eq;
this component creates the mapping between odd harmonics in Q and those in the
lightcurve. If _eq = 0, then an equatorial lightcurve of a uniform,
geometrically scattering asteroid will contain no odd harmonics. For this
reason, inversion of an opposition lightcurve furnishes an even-harmonic-only
version of Q called the symmetrized mean cross section, Qs-
Convex-profile inversion of the Cuyo lightcurves yields the estimates of
and Qs shown in Fig. i. These averages of Cuyo's shape rotate clockwise.
The Earth is at the top of the page and the Sun is _ degrees clockwise from
that direction. For each date, the rotation phase origin is defined by the
first data point on that date. A subtlety of Condition PHASE is that the _uLg]3
of # used in "convex-profile inversion" corresponds to the asteroid's rotation
sense. The sign is positive if the asteroid rotates through # from the Earth
direction to the Sun direction, a configuration that pertains to Cuyo in
October 1989 if the asteroid's rotat_on---_s dlrect. The inversions in Fig 1
that use a positive sign for _ are labeled "+DS" because here our view of the
mean cross section is from t-he south, i.e., pardilel to the spin vector. The
sign of _ is negative-if the asteroid rotates through _ fromt_e Sun direction
to the Earth direction, a configuration that pertains to Cuyo in October 1989
if the asteroid's rotation is retrograde. The inversions in Fig. 1 that use a
negative sign for _ are labeled "-RN" because here our view of the mean cross
section is from the north, i.e./ antiparal_e_ to the spin vector. For our
data, the two signs are equally acceptable and give nearly identical results.
w
z
OCT 7 +DS
' t t
Am 1 48
_* 1 13
Qc 4 00
OCT 8 +DS
I i
Am 1 50
_" 115
Oc 3 90
i
h
_ 180 360
ROTATIONAL PHASE
OCT 7 -RN
" f?c
1 48
1 13
4 00
OCT 8 -RN
180 360
ROTATIONAL PHASE
Arn
#.
f/c
I 50
I 15
3 90
Figure i. Cuyo lightcurves obtained on Oct. 7 and 8, and convex-profile
inversion results obtained for each date using +/- signs for the solar phase
angle (top/bottom piots). Estimates of Q and _s are shown as solid profiles
and dotted profiles. Lightcurves are plotted on _ scales, with unit
average brightness; the large symbols are the data, the solid curve was
derived from a Fourier series fit to the data, and the dotted curve was
derived from a constrained Fourier series corresponding to the estimate of _.
On the right, Am is the ratio Of the llghtcurve's maximum to its minimum;
_* is _'s breadth ra£io; and OciS Q's n0n-clrcuiarity, defined as the _
"distance" of the mean cross section from a circle (Ostro et al. 1988).
Asteroids, Comets, Meteors 1991 449
Cuyo's elongation
The mean cross section _ can be represented by a radius-of-curvature
function or by that function's Fourier series x. Under the ideal conditions,
there is a one-to-one mapping from the coefficients in x to the coefficients
in the Fourier series for the lightcurve. At phase angles as large as those
during the Cuyo photometry, Q's third harmonic (and to a lesser extent each of
its higher odd harmonics) propagates very efficiently into the lightcurve's
corresponding harmonic; see Fig. 3 of Ostro et al. 1988. That is why
convex-profile inversion of Cuyo's lightcurves, which contain a very strong
third harmonic, yields a "quasi-triangular" estimate of mean cross section.
The breadth ratio, _*, of Cuyo's mean cross section is estimated to be
about 1.14. That is, the elongation of the asteroid's mean latitudinal
contour is apparently very small. Here we have a good example of how the
amplitude of a large-_ lightcurve severely overestimates elongation.
This bias, which is commonly referred to as the phase-amplitude relation, is
a consequence of basic aspects of lightcurve-acquisition geometry that are
codified in convex-profile inversion theory (Ostro et al. 1988).
The estimates of Q and hence _* can contain systematic error if
Conditions GEO or EVIG are violated, as they must be to some extent). Ostro
et al. (1988) studied such error under simplified circumstances. At large _,
nongeometric scattering can distort estimates of Q, but the distortions are
not very severe. On the other hand, if the viewing/illumination geometry is
far from equatorial, then an elongated asteroid's _* can be severely
underestimated unless the Earth and Sun are opposite sides of the objects's
equatorial plane; see section IV.A of the referenced paper.
INFORMATION FROM RADAR
The Cuyo datasets from Arecibo and Goldstone (Ostro et al. 1991) consist
of echo spectra that provide fine rotation-phase coverage but are not quite
strong enough to yield a reliable estimate of the object's pole-on silhouette,
as has been done for a few other objects. However, visual inspection of the
spectra leads to two useful conclusions, as follows.
First, the echo bandwidth varies imperceptibly with rotation, supporting
the inference from the lightcurve inversion that this asteroid is not very
elongated. Second, the Goldstone and Arecibo weighted-sum spectra have
bandwidths (~160 Hz, ~40 Hz) only slightly larger than the ratio of the
telescopes' transmitter frequencies (8495 MHz, 2380 MHz), implying that cos 5,
the cosine of the asteroid-centered declination of the radar, did not change
very much during Sep 26 - Oct ii despite a 34 ° change in geocentric direction.
This result suggests that the view during the photometry was closer to
equatorial than to pole-on, and that the view may have been slightly closer to
equatorial during the October observations than in September. Hence, we find
no evidence in the radar spectra for severe violation of Condition EVIG.
CONCLUSION
Our photoelectric and radar observations support the hypothesis that
1917 Cuyo is a relatively unelongated asteroid.
ACKNOWLEDGMENTS
WZW gratefully acknowledges support by NASA Grant NAGW-716. Part of
this research was conducted at the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration (NASA).
450 Asteroids, Comets, Meteors 1991
ARECIBO SPECTRA FROM 45° GROUPS
64 °
60 0 -60Hz
DOPPLER FREQUENCY
ARECIBO SUM
r v "1
60 0 -60 Hz
DOPPLER FREQUENCY
GOLDSTONE SUM
-214 0 214 Hz
DOPPLER FREQUENCY
Figure 2. Radar echo spectra for 1917 Cuyo from observations reported by
Ostro et al. (1991). In each plot, echo power in the OC polarization is
plotted on an arbitrary linear scale vs. frequency, relative to the central
frequency estimated by those authors. A horizontal line is drawn at the zero
power level and a vertical bar at the origin indicates ± 1 standard deviation.
At the left are weighted sums of Arecibo spectra from eight, 45 ° intervals of
rotation phase; £he weighted-mean phase is given. On the right are Arecibo
and Goldstone weighted-sum spectra, smoothed to frequency resolutions (4.0,
14.3 Hz) and vignetted to windows (120, 428 Hz) in the same ratio as the
instruments' transmitter frequencies (2380, 8495 MHz). These spectra can be
thought of as one-dimensional images, or brightness scans across Cuyo's disc
through a slit parallel to the projected spin vector. The length e_ivalent,
in units of km/cos_:of frequency=is_the s&me for aii=the spectra in this
figure. _, the asteroid-centered deciination of the radar, may have changed
during the dates spanned by the radar observations; see text. The ~40-Hz
bandwidth Of ih6Arecibo sum, when co_ined wfth the rotation per_6d from the
ilghtcurve, implies that the maximum breadth of Cuy0's pole-on silhouette is
~3.9 km/cos _A, with _A the value of _ during the Arecibo runs.
:: REFERENCES
Harvey G. R. (1989) Close approach ephemerides of 1865 Cerberus and 1917 Cuyo.
Minor Planet Bull., 16, 36-37. :: _:
Ostro S. J., Connelly R., and Dorogi M. (1988) Convex-profile inversion of
asteroid lightcurves: Theory and applications. Icarus, 75, 30-63.
Ostro S. J., Campbell D. B., Chandler J. F., Shapiro I. I., Hire A. A.,
Velez R., Jurgens R. F., Rosema K. D., Winkler R., and Yeomans D. K. (1991)
Asteroid radar astrometry. _, 102, 1490-1502.


The shape of asteroid 1917 Cuyo

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