5 Rotation period of the nucleus

The period of rotation P of the nucleus was determined from position angles $\theta$ of the maximum elongations of near-nucleus isophotes. Selected images subject only to basic reduction and with the highest S/N were used. The inner isophotes of three April 21, 24 and 25 images are shown in Fig. 2.

  

Figure 2: Isophote contours in 81$\times$81 pixel (19$''\times$19'') fields, cropped from the images displayed in Fig. 1 (from left to right, April 21, 24 and 25), and aligned on the calculated photo-centre. Signal-to-noise (S/N) ratios are, respectively, $\sim$250, $\sim$260 and $\sim$90 with respect to the brightness of the photo-centre. Pixel intensity scale is linear and normalised to the interval [0, 1]. Isophote contours are drawn at intensity levels of 0.9, 0.8, 0.7, ... Cardinal points and the direction to the Sun are given. The upper scale bar indicates a spatial distance of 10000 km at the geocentric distance of Hale-Bopp, the lower bar an angular distance of 10''. The field size is $28\,000$ km. The abscissa and ordinate show relative pixel coordinates in the equatorial system (1 pixel = 0.24'')

The method consisted of identification of ($r_{\rm max}$,$\theta$)-coordinates. These defined the radius ($r_{\rm max}$) and position angle (PA) of the maximum distance from the photo-centre of a specific isophote. The isophote intensities were selected relative to the brightest pixel near the photo-centre. The data points of a segment of each generated isophote centered on the point of maximum elongation was then fitted by least squares with a parabolic segment, whose extreme point defined ($r_{\rm max}$,$\theta$) of the isophote data subset. Up to 13 isophotes per image were calculated, from which a fraction of 0.2 of the data points were generally selected for the fit. The extreme points obtained from isophote fitting were then used to produce a mean ($r_{\rm max}$, $\theta$)-profile, displaying the PA of the near-nucleus elongation as a function of distance from the photo-centre (Fig. 3) for the measured range of radii. This method was applied to each image.

  

Figure 3: Fitted isophote data for a number of coma intensity values, for an image obtained April 24 at 21:20.6 UT with a $\lambda$ 550 nm filter. Each isophote data set has been fitted with a second order curve segment, whose extreme point defines the maximum elongation, i.e. the radius and position angle of the point most distant from the photo-centre, of the particular isophote. The obtained e xtreme points of the data sets are indicated by asterisks. The extreme points have been fitted with another second order curve (nearly vertical), defining the ($r_{\rm max}$, $\theta$)-profile, giving the position angle of the near-nucleus elongation as a function of distance from the photo-centre. Ordinate scale is in pixels (1 pixel = 0.24'')

Based on images obtained within single nights, times of exposures and shapes of profiles allowed a first determination of the rotational profile P(r). The errors were largest at the inner and outer range of radii, due to the near-circularity of isophotes close to the photo-centre, and the smaller number of images with sufficient S/N at large radii. Combining single-night data from April 23 and 24 yielded a mean nucleus rotation period $P=11\hbox{$.\!\!^{\rm h}$}3\pm2\hbox{$.\!\!^{\rm h}$}7$ over a range of radii $r=600-9\,600\ \rm km$ from the photo-centre. The error bounds is for a 50% confidence level, as in the case of all the errors quoted in this work.

The error was reduced by assuming that the same filament, or jet complex, responsible for the major isophote elongation was stable and active throughout the full observing period. Taking into account the ($r_{\rm max}$, $\theta$)-profile data from all available nights, an improved P(r) was calculated, yielding a nucleus rotation period of $P=11\hbox{$.\!\!^{\rm h}$}46\pm0\hbox{$.\!\!^{\rm h}$}25$. This value is based on calculating the rotation period at zero radius, based on profiles which were fitted for radii $r=800-9\,300\ \rm km$, as described above. The available data points and rotation periods derived are shown in Fig. 5 for a number of radii.

The rotation of the nucleus was in the clockwise sense, thus the north pole of the nucleus was directed towards the Earth.