tokens: major and minor axes for a particular case. Since the model is specifically constructed in such a way, the asymptotic behaviours of the density are indeed given by whereas the Taylor–Maclaurin series is where L 3 = 3ln [( a + b )/ r 0 ] − 1 and L 15/8 = 15ln [( a + b )/ r 0 ] − 8. This last series also implies that the density profile might be increasing near the origin if ( a + b )/ r 0 is chosen too small, but the examinations of and indicate that r 0 ≤ ( a + b )e −8/15 ≈ 0.5866( a + b ) is actually sufficient to guarantee the positive non-increasing density profile. Figure 4. Same as Fig. 1 except for the density profiles in equation ( 51 ) with r 0 = ( a + b )/e 8/15 . Figure 5. Density profiles along the major and minor axes for the models in equations ( 45 ) and ( 51 ). For both cases, a = 2.58 and b = 0.43 but r 0 = e −1/3 ( a + b ) ≈ 2.157 for equation ( 45 ) and r 0 = e −8/15 ( a + b ) ≈ 1.766 for equation ( 51 ). 6 THE PROLATE AND TRIAXIAL SUBSTITUTIONS One possible generalization of the Miyamoto–Nagai substitution is applying a similar transformation to the other coordinates. For instance, consider the transformation of a spherical function Φ s ( r ) to Φ gMN ( R, z ) = Φ s ( t ) where t 2 = ( c + R d ) 2 + ( a + z b ) 2 and . With Φ gMN ( R, z ) as the potential, the resulting gravity field is given by with The gravitational force points locally to a more vertically inclined direction than that to the centre if a / z b > c / R d and vice versa, whilst it points radially towards the centre everywhere (i.e. the spherical potential) if a = c = 0. The equipotential in the limit of R, z → 0 is oblate if a / b > c / d or prolate if c / d > a / b , whereas the equipotentials as R, z → ∞ become spherical. If d = 0 and c ≠ 0, we also find is non-zero finite and so the gravitational force becomes undifferentiable along any path that crosses the symmetry axis. This behaviour is understood from the fact that the corresponding density profiles are singular in this limit (like ∼ R −1 ) on the R = 0 axis. The Poisson equation indicates that the axisymmetric density profile generating the potential Φ s ( t ) can be expressed as generalizing equation ( 15 ), which corresponds to the c = d = 0 case. It is easy to observe that this density profile is also always non-negative if (actually ρ s ( r ) being non-negative and integrable at r = 0 is sufficient), and, from explicit calculation, we can furthermore demonstrate that and if (ρ s )′( r ) ≤ 0 (and ). Since , equation ( 56 ) is also regular everywhere with the Taylor-MacLaurin series unless bd = 0. 5 Here, If d = 0, on the other hand, equation ( 56 ) reduces to which exhibits an explicit R −1 -density singularity as R → 0 (i.e. the density along the symmetry axis diverges). This contrasts to the b = 0 case, for which a razor-thin disc develops. 6.1 The Miyamoto–Nagai spindle The simplest potential-density pair that results from this transformation is that due to the point mass potential Φ s ( r ) =
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 B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-URL I-URL I-URL I-URL I-URL I-URL I-URL I-URL I-URL I-URL I-URL I-URL I-URL I-URL O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula