I think there's an easier way to convert .

1. For the positions :

The scaling factor is : 778547200000 ( ie = sma.Jup ) , while in the paper the sma = 1

Xsun = - mu * SmaJup = -mu * 778547200000 =-743350581.5

XJup = (1-mu) * SmaJup = (1-mu) * 778547200000 = 7.77804E+11

Xast = Xgiven * 778547200000 = -8.26825E+11

2. For the velocities :

The velocity of Jup is given by : v Jup = sqrt ( G*Mt * (1-mu)^2 / sma ) = 13051.22627

The velocity of Sun is given by : vSun = - vJup * Mjup/Msun = - 12.46120548

The scaling factor for the velocity is = 13063.69939.

This value is obtained by using the above formula once with GMt=1 and sma =1 ; and the second time with the real values of GMt and Sma.

So the velocity of the asteroid is : V ast = V given * 13063.69939 = 1417.54202

Please note the above values are referenced to the center of mass , which has zero velocity .

Edit : when plugging in the above values I see I don't get the desired orbits . Maybe the velocity parameter ( 0.10851) is described in a rotating frame , as a would have expected a value close to 1 . I think I must check this out