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 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 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 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 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 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 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 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 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 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 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 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 O O O O O O O O O O O O
tokens: ). Stars and SPH particles were coupled by direct point-to-point interaction, without any approximation for the gravitational field. Finally, we corrected the gas positions and velocity according to Eq. ( A.4 ) by recalculating the accelerations and the energy rates at the new stage n + 1. Appendix B: Technical features B.1. Neighbor search and acceleration updating Before we employed it to compute gravitational interactions, the tree grid was also used to support the operations related to the nearest-neighbor search. To calculate the density and the smoothing length, each i th particle starts with a first-guess value of h i and makes a tree walk by adopting an opening criterion that is slightly different from Eqs. ( 20 ) and ( 21 ). Given a generic particle of index i, in order to find the other points enclosed in its SPH kernel support, we checked for each cube the overlap with a sphere of radius 2 h i centered onto the point r i = ( x i , y i , z i ). A box is opened only if the following three conditions are valid: where ( x A , y A , z A ) are the coordinates of the geometrical center of the cube and D L is its side length. The tree walk continues by opening the further cubes according to the last rule, until the single particles are reached and the neighborhood is thus determined. At the end of the walk, a temporary value of density ρ i is calculated. If the number of encountered neighbors differs from the expected number (and the difference exceeds the tolerance number), h i is updated according to Eq. ( 5 ). Then, a new tree walk is performed, and a new value for the density is computed. The tree walk and the h i , ρ i updating are executed cyclically until the number of neighbors converges on the desired value. During the tree walk, the various quantities P i , Ω i , ζ i , f , ∇ ⋅ v i , and ∇ × v i are computed at the same time together with the density. After this preliminary phase, the acceleration a and the thermal energy rate can be computed by performing a further tree walk for each particle. During the walk, the hydrodynamics contribution to the acceleration and the gravity field are computed at the same time. For each box, before Eq. ( 20 ) (or Eq. ( 21 )) is applied, the code primarily checks Eq. ( B.1 ) to distinguish cubes that might contain some neighbor particles for the SPH interpolation. If this is satisfied, the box is opened, otherwise the particles inside will not contribute at all to hydrodynamics. After this primary check, the algorithm performs a secondary control by applying the opening criterion of Eq. ( 20 ) (or Eq. ( 21 )) to decide whether the multipole approximation can be applied to the Newtonian field. As a result, practically, a generic i th particle will interact with its local SPH neighborhood following Eqs. ( 7 ) and ( 8 ), experiencing the Newtonian force through a direct particle-to-particle interaction. The remaining particles that lie outside the SPH domain contribute to the accelerations with or