ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: of sight is chosen to be along the x- axis, it is useful to express the x positions of each surface with a specific r rc asDefining a new functionyields denotes the two line-of-sight ( x ) positions for a surface defined by a particular value of r rc at sky position ( y , z ). Functions a , b ( y , z ), and c ( y , z ) are defined by the geometry of the object in question. Equations ( 6 )–( 8 ) describe the appropriate functions representing a spheroid with axial ratio α and inclination i. E ( y , z ) is a function that is entirely dependent on the object's shape; the following section discusses its conceptual meaning further. 3.1. Discrete Derivation It is possible to define an object as a discrete series of shells, each of which is defined as the region between an inner () and an outer () surface with an average volume density () within the shell. The depth along the x- axis of each such shell at different ( y , z ) positions will vary according towhere is the total depth along the line of sight at position ( y , z ) for the shell made up of two surfaces defined by and . Substituting Equation ( 11 ) into Equation ( 12 ) yieldsEquation ( 13 ) describes the depth of each shell; however, this representation is of limited use since r rc is not an observable quantity. Similarly, E ( y , z ) is a function that is directly dependent on the object's unknown shape. Equation ( 13 ) must be put in terms of observable quantities: the observed column density, and the area within each column density contour. Each surface as described by Equation ( 11 ), when projected onto the line of sight, produces a closed boundary composed of those ( y , z ) positions where . In view of Equation ( 11 ) the projected boundaries of each shell are defined bySolving for E ( y , z ) using the spheroid model above yields a familiar relation,which is a simple ellipse that results from projecting a three-dimensional spheroid surface onto a two-dimensional plane. Equation ( 14 ) makes clear that each contour of equal E ( y , z ) corresponds to the boundary of a particular volume density surface, with a specific value of r rc , and possesses a unique projected area. In general, the projected area of a surface of a particular r rc can be expressed aswhere is a geometry-dependent unknown constant(). All positions with equal correspond to the projected boundary of a surface with with corresponding area A c . The additional subscript c denotes a specific contour. Therefore, Equation ( 13 ) can be reformed in terms of areas aswhere represents the depth along the line of sight of the shell between surfaces defined by and at all positions defined by the contour formed by the projected boundary of the surface defined by . The observed column density can then be defined as N c represents the column density at all positions ( y , z ) defined by the projected boundary of the surface. represents the mean volume density within the shell whose surfaces are defined by and . The column density