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TABLE I.—Coordinates of the Rotation Axes for Selected Asteroids
Asteroid Ecliptic longitude, Ecliptic latitude, Right ascension, Declination, Obliquity Reference
6 . . . . . . . . . . . . . . 145 15 153 27 77 Gehrels and Owings (1962)
184 55 212 47 30 || Cailliatte (1956)
348 76 299 60 9 || Chang and Chang (1962)
115+10 19+5 121 40 64 || Sather and Taylor (1972)*
103 61 141 82 21 Cailliatte (1960)
105 30 112 52 58 Gehrels and Owings (1962)
122 10 127 29 84 Gehrels and Owings (1967)
*Denotes poles determined by photometric astrometry. TABLE II.-Pole Coordinates of Asteroid 433 Eros
Reference No 80 do 60
Zessewitsche (1937) 29° 22° 18° 31° Rosenhagen (1932) 4 45 342 42 Watson (1937) 349 62 316 51 Krug and Schrutka
Rechtenstamm (1936) 2 53 333 48 Stobbe (1940) 9 38 350 38 Beyer (1953) 353 13 349 9 Cailliatte (1956) 10 46 345 45 Vesely (this paper) 13 28 0 31
and declination d of the Earth. Plotting the observed amplitude A against d, Zessewitsche derived the following empirical relationship:
where 1.50 mag is the maximum observed amplitude. We refer to this as an amplitude-aspect relationship. Zessewitsche's method relies on a large number of transformation equations. The value for the inclination of the equator is an important parameter in several of them. Zessewitsche admitted a lack of precision in its calculation. M. Huruhata (1940) misinterpreted Zessewitsche's pole value and his work must be viewed with suspicion.
Graphic Pole Determinations
F. Watson (1937) plotted the observed amplitude as a function of the ecliptic coordinates of Eros during an opposition. Assuming that the greatest amplitude was observed when Earth was in the equatorial plane of Eros, Watson used a composite of the curves of oppositions from 1893 to 1935 to secure the coordinates of the nodes of the equatorial plane of Eros and its inclination, from which he determined the pole. Watson used published amplitudes from various oppositions, combining photometric and photographic data without indicating the probable error of the observed magnitudes and included data regarded as imprecise by Stobbe (1940). Checking the original published lightcurves, I found variations between different observers on the same night to be as high as +0.3 mag, and I completely reject his pole. J. Rosenhagen obtained an approximate pole from a plot of the amplitude against the Earth's ecliptic coordinates. The refinement will be discussed under “Mathematical Models for Pole Determinations.” J. Stobbe (1940) used Zessewitsche's amplitude-aspect relationship to determine da, the Erocentric latitude of Earth from the observed amplitude. With various assumed pole positions, he calculated a series of dr values (B = beobachtet; R = rechnet). Using the de values as normal points, curves of the dR values for the various assumed pole values were plotted for the dates of observation. The results were indeterminate, so he used another relationship as follows. By using a period calculated from the previously approximated pole, Stobbe found (amin’s)a, the observed variation in the time of arrival of the minimum due to change in phase. Using the assumed pole positions, he determined (amin’s)R, the calculated variation. The resulting B-R plot, along with the do plot, enabled Stobbe to elect the best assumed pole from pole curves. To determine possible agreement between his possible poles for the 1930-31 opposition and van den Bos and Finsen's (1931) position angles, Stobbe plotted his pole coordinates as points and drew the great circle corresponding to the position angle of the pole. He presumed the intersection of the points and the great circle to be the pole of Eros. The result agreed with one of his values, but not the one he felt best represented the pole of Eros for the opposition 1930-31. Stobbe indicated the disagreement was possibly due to irregularity of the figure or flexure along the long axis and claimed his findings vindicated the often skeptically received (Stobbe, 1940) observations of van den Bos and Finsen (1931). Although Stobbe selected one pole as most satisfying, we see he has as many poles as he has oppositions. He claimed the pole is not fixed. On the contrary, I believe the slope of an amplitude-aspect plot determined for observations at a certain phase angle and obliquity will be valid for that opposition only and will yield a different pole for another opposition unless the conditions are the same. Stobbe rejected obviously unsure observational data, but the ones he accepted may not be accurate. The inability to secure a single pole may indicate systematic error of the method.
Mathematical Models for Pole Determinations
J. Rosenhagen (1932) showed that his graphically approximated pole (see above) must be refined in terms of Eros' shape. He assumed an elongated body rotating about the short axis, similar to a Poincaré body or symmetrical egg figure, with a brightness proportional to the projected area when viewed equatorially. To yield an Eros maximum amplitude of 1.50 mag required an axial ratio of a;b = 4.00 and an eccentricity of e = 0.97. Rosenhagen devised an amplitude-aspect relationship based on this model. Starting with the approximate pole, he made differential corrections until he determined the pole yielding the aspect that best conformed to the requirements of the model.
Rosenhagen found that his pole gave the right amplitude for the observations of 1930-31, but would give a maximum amplitude of only 1.14 mag for the earlier oppositions (1901-1903). He blamed systematic deviations of the data due to precession, deformation, and spotting of the asteroid. Rosenhagen's pole may be challenged for more obvious reasons. He tried to intercompare amplitudes among oppositions whose observational data produced a range of phase coefficients from 0.011 to 0.039 mag/deg and included data that Watson (1937) said was not comparable because of uncertainties in the magnitudes of the comparison stars. Rosenhagen's pole generates little confidence. W. Krug and G. Schrutka-Rechtenstamm (1936) proposed to determine the brightness of a three-axis ellipsoid model of Eros at full phase while obeying Lambert's law, using only photometric observational data having both amplitude and absolute magnitude reduced to an average opposition. They related absolute brightness to the aspect angle, which was determined from Rosenhagen's pole. A least-squares solution gave a corrected pole. Krug's new pole met the brightness conditions required by the model, but did not permit sufficient maximum amplitude. Because Krug and Schrutka-Rechtenstamm's pole would not permit maximum amplitude, and they used data from observers common to Rosenhagen, criticized previously, this pole, too, must be considered very doubtful. F. E. Roach and L. G. Stoddard (1938) revised the work of Krug and Schrutka-Rechtenstamm. They assumed Krug's pole, but related the brightness ratio to the maximum amplitude, omitted observations with no variation, and gave no weight to absolute magnitude. Their least-squares solution allowed for a maximum amplitude of 1.50 mag for Eros. Thus, using old, imprecise photometric data and their own, single photoelectric lightcurve, Stoddard and Roach perhaps improved a model, but shed no further light on Eros' pole.
MORE RECENT POLE DETERMINATIONS
M. Beyer (1953) used Stobbe's (1940) method for his determination of the pole of Eros. Although his observational data are more precise, the pole seems to be unreasonably low.
Cailliatte (1956) used the geocentric coordinates of an asteroid for two observations of the maximum amplitude to determine the longitude of the node and the inclination of the equator of the asteroid. The pole thus determined was used to calculate D, the asterocentric declination of Earth. Cailliatte then plotted an amplitude-aspect relation that he refined using various models. He used the refined amplitude-aspect relationship to correct the original pole. In a later publication (1960) he corrected two earlier poles. Cailliatte's amplitude-dependent method required larger than observed maximum amplitudes for some asteroids (e.g., for 39 Laetitia: 0.68 mag calculated, 0.54 mag observed) and yields generally small obliquities.
Y. C. Chang and C. S. Chang (1962, 1963) determined a number of poles using an amplitude-aspect relationship. They used a single reduced observation and the asteroid's phase coefficient as the factor by which the amplitude varies with D, citing Cailliatte (1956) as the source. Actually, Cailliatte indicated this was a “restrictive hypothesis,” somewhat better than no method at all. The Chang poles have no value.
P. Tempesti and R. Burchi (1969) also made use of an amplitude-aspect relationship:
where A, is one of 12 observed amplitudes, A0 is an assumed maximum (unknown), C is a constant (unknown), and d is the asterocentric declination of Earth. They used each A; and increasing values of C with each assumed A0. A least-squares solution analyzing the relative minima of residuals indicated A0 = 1.50 mag and C= 0.0146 mag-deg yielded minimal standard error. They transformed the value received for d into pole coordinates. Tempesti and Burchi stated the error may be large because of the small range of amplitudes. Greater faults appear evident. There is no observational justification of a maximum amplitude of 1.50 mag. Also, a partial lightcurve (4% hr) of April 6, 1970, gives an amplitude of 0.07 mag at a time when, according to Tempesti, a nearly equatorial view was anticipated.
The early poles of T. Gehrels and D. Owings (1962) were determined using an amplitude-aspect relationship:
where a is the observed amplitude, A the greatest possible amplitude, and Y the angle between the direction of observation and the axis of rotation. A sin Y master curves were made for different values of A and 60 plotted as a function of (A-\o). 3 = 0 was assumed; and when only two observations were available, it was assumed 60 = 8°. The master curves were superposed on longitude plots of a 1, a2, and the visual absolute magnitude. A weighted average was given for the determined longitude No, giving half-weight to the absolute magnitude and to poor determinations. The latitude of the pole was determined from the quality of fit to the observations by the different sets of master curves. Gehrels claims little precision for the latitudes and no determination of a sign. The observational data are good and the phase angles were usually small. The pole longitudes are more precise, as they do not depend strongly on the assumed amplitude-aspect function. Recognizing the unreliability of the amplitude-aspect relationship, Gehrels (1967) developed the photometric astrometry method described by R. Taylor! and determined the pole of 4 Vesta in 1967. An error in cycle correction discovered later causes us to lack confidence in this determination. We believe the pole is within +10° of the published coordinates.