1D protostellar disk sub-grid model for star formation 3D MHD simulations

Author: Anaïs Pauchet

Intro: Massive star formation is still poorly understood and the rarity of their events as well as their embedded nature make this mechanism really difficult to observe. Simulations have tried to bypass these difficulties but one major issue remains: star formation spans a large range of spacial scales which leads to the impossibility of resolving stars with a reasonable amount of numerical resources. They are instead described as sink particles (SP), point like objects accreting mass.

Objective: My PhD project aims to improve the description of the SP inside 3D MHD simulations of dense core collapse. So far, in these simulations each SP represents one single star without accretion disc. This is the reason why, I am developing a 1D protostellar disc model which will be included as a subgrid model in FLASH.

Method: For the last two years I have been deriving the equations of a viscous disc to follow the evolution of the system disc+star, I implemented a numerical model in python to post process the evolution of SPs in existing FLASH simulations made by Birka (Zimmermann et al. 2025).
As my model is supposed to be part of the 3D simulations, I need to make sure that all the quantities are conserved, mainly the mass and angular momentum of the SP. To not increase the computational complexity in the 3D simulation, the model is relatively simple. This is why we had to make some assumptions. For example we kept the rotation velocity of the disk constant, following a Keplerian profile, observed on numerous disks. However, some mechanisms could make this assumption invalid, like:
- The Keplerian profile depends on the mass of the star. The higher the mass of the star, the higher the rotation velocity.
- The direction of the angular momentum of the accreting matter can influence the rotation of the disk. Matter accreting in the opposite direction of the disk rotation slows down the disk.
We decided to take care of those mechanisms through the disk flux. Instead of accelerating/decelerating the disk we add a term in the flux equation which influences the transport of matter.
For example, for a Keplerian profile, the mass of the star will increase which will lead to the acceleration of the disk (Ω∝M_⋆^1/2), but additionally the star being bigger its gravity will be higher and matter will more rapidly move inward. A term which is rarely taken into account in disk models because the mass growth is generally negligible.
On the other hand for an accreting angular momentum orthogonal to the disk rotation, the matter should be slowed down (in 3D simulations) and more rapidly fall onto the star. So we add a term in the flux that describes that without slowing down the disk.

Preliminary results: In Figure 1 we can see that the SP (green) mass and angular momentum correspond to the total mass and angular momentum of the disk model. We can then say that those two quantities are conserved. We can notice that when the angular momentum is decreasing, the accretion onto the star increases drastically, mass is flushed from the inner disk to the star.
Figure 2 displays the evolution of the disk over time and we can see that without modifying the velocity of the disk, when the accreted angular momentum is negative, the inner disk is empty because its matter fell rapidly onto the star.