Equation of the field lines of an axisymmetric multipole with a source surface
(Color online) A star of radius , shaded gray, with an axial dipole magnetic field with a source surface (the large dashed circle) of radius . The source surface allows for the inclusion of regions of open field lines, along which a stellar wind is launched, in addition to the regions of closed field lines. Each closed field region is surrounded by regions of open field lines. The field lines are shown in light gray. The solid black lines illustrate particular field lines, with their footpoints on the stellar surface denoted by the solid black circles. Open field lines have a single footpoint, and closed field lines have two footpoints on the star. The field lines within each region of closed field lines reach their maximum radial extent of along a line of constant polar angle. Different values of correspond to different field lines within each closed field region. The long-dashed line denotes the closed field line loop with . The footpoints of this closed field line represent the boundary at the stellar surface between regions of closed and open field.
(Color online) (a) A flattened polar projection showing the radial field component of the surface magnetic field of the forming star V2129 Oph (Ref. 15). The stellar equator is shown as the bold circle, with lines of constant latitude separated by 30° as the dashed lines. There is a positive field spot slightly offset from the pole, surrounded by a ring of negative field, which is surrounded by a ring of positive field below the equator. Fluxes are given in Gauss. Numbers and tick marks around the circumference denote the rotation phase and phases of observation respectively. The large scale magnetic field is found to be dominantly octupolar, which is apparent from the (b) extrapolation of the three-dimensional coronal magnetic field, which is constructed from the magnetic map. Closed field lines are shown in white, with open field lines, along which a stellar wind could be launched, shown in blue (see the online version for colors). The large scale field approximately resembles a tilted dipole, and the medium scale field resembles an octupole with three rings of closed field. Details of the observational techniques used to construct stellar magnetic maps and the numerical field extrapolation model can be found in Refs. 11 and 16, respectively.
(Color online) The first three lowest order multipoles, the dipole , the quadrupole , and the octupole . The multipole moment symmetry axis, denoted by for the dipole, for the quadrupole, and for octupole, is assumed to be aligned with the stellar rotation axis, denoted by . The plus/minus signs denote regions of positive/negative field. The order of a multipole is the number of polarity changes in the surface field between the north and south pole of the star along meridians (lines of constant longitude); is the number of roots of between and . A multipole of order has regions of closed loops around the entire star in each meridional plane (planes with ). In three dimensions the closed field regions form rings of closed field around the star, with rings for a multipole of order .
A field vector decomposed into the radial and polar components at a point along a field line a distance from the center of the star (shaded gray) at a co-latitude of . The field components are used to illustrate their definitions and are not to scale. The field line reaches a maximum radial extent of . Only the first quadrant is shown because the magnetic fields considered in this paper are reflectionally symmetric in the horizontal axis and rotationally symmetric about the vertical axis.
(a) The field lines of a dipole , (b) a quadrupole , and (c) an octupole with a source surface at plotted as the dashed line. The star is shaded in gray. The shapes of the closed field lines are calculated for each multipole from Eq. (31). The shapes of the open field lines are calculated in a similar way from to by considering the range of co-latitudes not occupied by the footpoints of the closed loops. For the quadrupole and the octupole (and other higher order multipoles) there are regions of open field at lower latitudes along which a stellar wind could be launched. The magnetic fields are rotationally symmetric about the -axis and reflectionally symmetric in the plane.
(Color online) A star (shaded gray) with an axial hexadecapole magnetic field . denotes where the closed field lines reach their maximum radial extent, and indicates the locations of the footpoints of the closed loops. For clarity only two closed field line loops within each closed field region are shown (solid lines), and no open field lines are shown. The field is rotationally symmetric about the -axis and reflectionally symmetric about the -axis, and therefore we need consider values only between zero and . The field is purely radial at the stellar rotation pole, in the equatorial plane, and along a line between the closed field lines shown here (and by symmetry in the other three quadrants). Because and , the values of where and are zero correspond to the roots of the particular Legendre polynomial and associated Legendre function, respectively.
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