Numerical and analytical calculations demonstrate that small planetesimals exhibit long-term stability of libration about the L4 and L5 Lagrange equilibrium points at a proto-Jupiter's distance from the Sun despite continuous dissipation of energy due to gas drag in the primordial solar nebula. This stability may mean that most of the mass of the Trojan asteroids was collisionally captured into libration and accreted into large bodies while the solar nebula was still there. If this is so, the Trojans represent samples of the material that composed Jupiter's core before the addition of the large mass of volatiles leading to the current largely gaseous planet.
For a (13-M_{\oplus}) Jupiter core that is assumed not to disturb the mass distribution in the solar nebula, dissipation of orbital energy while the planetesimal is further from the Sun than the stationary point in the frame rotating with Jupiter (or the elliptic fixed point in a surface of section if Jupiter's orbit is eccentric) decreases the amplitude of libration. However, dissipation while the planetesimal is closer to the Sun than this point increases the amplitude of libration. The effects of dissipation outside the stationary point nearly cancel the effects of the dissipation inside, resulting in either a very slow increase or decrease in the amplitude of libration depending on the details of the nebular model. Those models of the nebula leading to a larger ratio of drag forces equidistant outside and inside of Jupiter's orbital distance give the librating planetesimals a higher stability. By higher stability we mean that the planetesimal is more likely to remain permanently in the region with decreasing amplitude of libration, or if the amplitude is increasing, that the rate of increase is lower. The L4 stationary points of Trojan precursors experiencing nebular drag are located more than (60^{\circ}) in front of Jupiter, whereas the L5 stationary points are located less than (60^{\circ}) behind Jupiter. The separation of these stationary points from (\pm 60^{\circ}) increases as the particle diameter (d) decreases or the drag acceleration is otherwise increased. For a (\mathbf{1 3}-M_{\oplus}) Jupiter and a nominal nebular model the stationary points at the L4 equivalent reach a maximum angular separation of about (108^{\circ}) in front of Jupiter for (d \approx 30 \mathrm{~m}), where smaller planetesimals do not librate in front of Jupiter. In contrast, the stationary points may be located only a few degrees behind Jupiter for (d \approx 1 \mathrm{~m}). If