We consider in detail the situation of applying a time dependent external

magnetic field to a 87Rb atomic Bose-Einstein condensate held in a harmonic

trap, in order to adiabatically sweep the interatomic interactions across a

Feshbach resonance to produce diatomic molecules. To this end, we introduce a

minimal two-body Hamiltonian depending on just five measurable parameters of a

Feshbach resonance, which accurately determines all low energy binary

scattering observables, in particular, the molecular conversion efficiency of

just two atoms. Based on this description of the microscopic collision

phenomena, we use the many-body theory of T. Koehler and K. Burnett [Phys. Rev.

A 65, 033601 (2002)] to study the efficiency of the association of molecules in

a 87Rb Bose-Einstein condensate during a linear passage of the magnetic field

strength across the 100 mT Feshbach resonance. We explore different,

experimentally accessible, parameter regimes, and compare the predictions of

Landau-Zener, configuration interaction, and two level mean field calculations

with those of the microscopic many-body approach. Our comparative studies

reveal a remarkable insensitivity of the molecular conversion efficiency with

respect to both the details of the microscopic binary collision physics and the

coherent nature of the Bose-Einstein condensed gas, provided that the magnetic

field strength is varied linearly. We provide the reasons for this universality

of the molecular production achieved by linear ramps of the magnetic field

strength, and identify the Landau-Zener coefficient determined by F.H. Mies et

al. [Phys. Rev. A 61, 022721 (2000)] as the main parameter that controls the

efficiency.