The solar corona is composed of two components:
We present below a novel approach to the problem. We first construct a Time Intensity Diagram (TID) by piling circular profiles extracted from a time-series of images at a given radial distance. Incidentally, the origin of the polar coordinates is not at the center of the Sun but closer to the limb, at the two points (either north or south) of apparent divergence of the plumes. The TID is therefore an image giving the radiance as a function of position angle (x-axis) and time (y-axis) where plumes appear as bright points. Coherent alignments, that is trajectories, are first visually detected and then established using the Hough transform technique. The dotted pseudo-sinusoidal tracks allow to conclude that plumes are recurrent structures, transitory lit, in rigid body rotation with the solar corona.
The process of constructing the TID starts with a selection of a set of LASCO/C2 images covering at least half a solar rotation (14 days) with a minimum frequency of 2 images per day to insure interframe continuity. This set is rigorously photometrically equalized by comparing two successive images and correcting for any difference in radiance of more than 0.1 percent. A minimum background image is calculated from the full set after a median smoothing. Then for each equalized image of plumes the following steps are implemented:
Under the assumption of rigid body rotation, individual plumes will follow sinusoidal curves on the TID with a common axis corresponding to the polar direction. The instantaneous position angle can be written as a function of time. The maximum elongation (or colatitude), the period and the initial phase angle are the three parameters which define a particular trajectory. The exact equation, including a corrective term for the apparent variation of the polar axis with respect to the plane of the sky can be found in Lamy et al. (1998).
The TIDs built from white light images do not display continuous tracks for the plumes, except at high colatitudes. Their patchy appearance immediately suggests that identified plumes are not permanent structures. However dotted sinusoidal paths are suggested and must further be explored. Their reality will indicate that plumes are enduring, recurrent structures that are transiently lit.
In order to find objective tracks, we introduce a variant of the Hough transform, a method derived from the Radon transform, which makes it possible to detect many kinds of geometric alignments in an image. The specific advantage of the Hough method is its ability to detect alignments of disjoint tracks and points as a single entity. A discussion of this method can be found in Ballester (1994, 1996) and astrophysical applications, for instance in Ragazzoni & Barbieri (1993, 1994).
In our case, alignments would coincide with the sinusoidal tracks of the plumes. The method consists in summing the TID levels along the expected trajectory for each combination of colatitude, initial phase and period. In theory, the track profile over the TID is a Dirac function, but for practical situations the Dirac kernel is replaced by a so-called influence curve. It must be positive, integrable and with a finite support. We chose a normalized truncated gaussian curve corresponding to the estimated uncertainty in the plumes position over time.
Each sum defines a point in the 3-D parameter space. The result is a 3-D intensity function of the synodic rotation period which ranges from 27 days at the equator to 35 days at the poles, with a half day uncertainty, the colatitude which varies from 0 to 80 with a reasonable resolution of 2 and the phase which covers the whole period, with a half day resolution.
For an initial estimation, this function was reduced to a more tractable 2-D function, by fixing the period to the mean synodic solar rotation. The standard technique consists in determining first the coordinates of the absolute maximum, which defines the most conspicuous track, then to subtract from the TID all the points corresponding to the track and then to restart the whole process and look for the next maximum. In practise, we do not follow this procedure because the same point of TID can belong to two distinct trajectories. Instead, we select the most prominent set of local maxima at once as indicators of most probable trajectories.
We built the TID corresponding to a time interval of 43 days centered on December 1997, as shown in Fig. 1. We found 7 reliable trajectories with a period of 27 days and with colatitudes ranging from 14 to 54; five of them are overplotted in the figure. This method has been effective to track the dotted trajectories of the white light polar plumes establishing for the first time their temporal evolution and their correlation with the extreme ultra-violet plumes present in the solar polar regions (Llebaria et al. 1998).
Ballester, P. 1994, A&A, 286, 1011
, 1996, Vistas in Astronomy 40, 479
Lamy, P., Llebaria, A., Koutchmy, S., Reynet, P., Molodensky, M., Howard, R., Schwenn, R., & Simnett, G. 1997, in Proc. of the Fifth SOHO Workshop (ESA Publ. SP-404), ed. A. Wilson (Noordwijk: ESA), 487
Llebaria, A., Lamy, P., Deforest, C. E., & Koutchmy S. 1998, in Solar Jets and Coronal Holes (ESA Publ. SP-421), ed. P. Martens & T. D. Guyenne (Noordwijk: ESA), 87
Ragazzoni, R. & Barbieri C. 1993, in Proc. of Applications of Time Series in Astronomy and Metereology, ed. O. Lessi (Padova: Univ. of Padova), 233
, 1994, PASP, 106, 683