Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

226 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
of the radial grid in such a way as to satisfy the Jeans mass
and the three Jeans length conditions
for spherical coordinates, and still obtained fragmentation into a welldefined
binary. Only when the spatial resolution was increased to
and the solution changed behaviour [17]. That is, instead
of fragmenting into a binary, the central collapse produced a bar
which thereafter condensed into a thin filament in good agreement with
the AMR results of Ref. [10]. Thus, it appears that for grids based on
spherical coordinates, the four Jeans conditions (1) and (2) are necessary
but not sufficient for realistic fragmentation, and that these must be
combined with sufficiently high spatial resolution to ensure a converged
solution. In Ref. [17] it is argued that one reason for requiring more numerical
resolution in spherical coordinate calculations is the occurrence
of high-aspect ratio cells near the center, where and
may become large compared with a locally uniform Cartesian grid with
unit-aspect ratio.
Here we have recalculated the isothermal Gaussian cloud model using
a modified version of the spherical coordinate code described in
Ref [18] . In order to maintain a numerical resolution higher than that
demanded by the Jeans conditions (1) and (2), the gravitohydrodynamic
equations are solved in the zooming coordinates, defined by [19]
where denotes the isothermal speed of sound and the time difference
is in units of the central free-fall time
With these transformations, the gravitohydrodynamic equations can
be written as