Pair-instability mass loss for top-down compact object mass calculations



Introduction
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Stellar and binary population synthesis calculations are necessary to predict event rates and population statistics of astrophysical phenomena, including those involving neutron stars (NS) and black holes (BH). Typically, at the end of the evolution (carbon depletion) the mass of the core is mapped to a compact object mass, using a $M_\mathrm{comp.\ obj}\equiv M_\mathrm{comp.\ obj}(M_\mathrm{core})$ informed by core-collapse (CC) simulations (e.g., @fryer:12 [@spera:15; @mandel:20; @couch:20], see also @zapartas:21 [@patton:21]) and/or (pulsational) pair instability (PPI) simulations (e.g., @belczynski:16 [@woosley:17; @spera:17; @stevenson:19; @marchant:19; @farmer:19; @breivik:20; @renzo:20csm; @costa:21]).

The most commonly adopted algorithms to obtain compact object masses in the CC regime are the "rapid" and "delayed" prescriptions of @fryer:12. In both cases, the compact object is built from the "bottom-up", starting from a proto-NS mass and adding the amount of fallback expected in the (possibly failed) explosion. However the proto-NS mass and information about the core structure relevant to decide the fallback are usually not available in population synthesis calculations [e.g., @patton:20]. Instead the total final mass of the star is arguably easier to constrain in population synthesis models.

At the transition between CC and PPI (roughly at carbon-oxygen cores of $\sim{}35\,M_\odot$, @woosley:17 [@marchant:19; @farmer:19; @renzo:20csm; @costa:21]), a mismatch between commonly adopted fitting formulae exists. These are at least partly caused by differences in the stellar structures used to design these algorithms and impair the interpretation of gravitational-wave data [as pointed out in Fig. 5 of @vanson:21]. While it is possible that the BH mass function is discontinuous at the onset of the PPI regime (e.g., @renzo:20conv [@costa:21], Hendriks et al., in prep.), the location and amplitude of a putative discontinuity should not be governed by a mismatch between the fitting formulae from different stellar evolution models.

Top-down compact object masses
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In contrast with the "bottom up" approach of @fryer:12, we propose a "top-down" way to build the compact object masses $M_\mathrm{comp.\ obj}$. Starting from the total stellar mass we remove the amount of mass lost due to all of the processes associated with the (possibly failed) explosion:

$$\label{eq:mass}
      M_\mathrm{comp.\ obj} =
      M_\mathrm{pre-CC} - \left(\Delta M_\mathrm{SN} + \Delta M_{\nu, \mathrm{core}} + \Delta M_\mathrm{env} + \Delta M_\mathrm{PPI} + \cdots \right)$$

where all masses are in $M_\odot$ units, $M_\mathrm{pre-CC}$ is the total mass at the onset of CC, and each term in the parenthesis corresponds to a potential mass-loss mechanism: $\Delta M_\mathrm{SN}$ for the CC ejecta, $\Delta M_{\nu, \mathrm{core}}$ the change in gravitational mass of the core due to the neutrino losses, $\Delta M_\mathrm{env}$ the loss of the envelope due to the change in gravitational mass corresponding to $\Delta M_{\nu, \mathrm{core}}$ that can occur even in red supergiant "failed" core-collapse [@nadezhin:80; @lovegrove:13; @piro:13; @fernandez:18; @ivanov:21], and $\Delta M_\mathrm{PPI}$ the pulsational mass loss due to pair-instability. Each term may further be a function of the progenitor stellar or binary properties, and may be theoretically or observationally informed (e.g., $\Delta M_\mathrm{SN}$ could be derived from the light curves of a large sample of observed SNe). can be further extended by adding additional mass-loss mechanisms in the parenthesis (e.g., disk winds).

In the CC regime the previous approach from @fryer:12 can be recovered by setting $\Delta M_\mathrm{SN} + \Delta M_{\nu, \mathrm{core}} = M_\mathrm{pre-CC} - M_\mathrm{comp.\ obj}^\mathrm{Fryer+12}$, where the last term is the compact object mass as predicted by @fryer:12 and ignoring the other mass loss terms, such as $\Delta M_\mathrm{PPI}$ and $\Delta M_\mathrm{env}$.

New fit for PPI ejecta
======================

![Each panel shows our fitting formula for the amount of PPI induced mass-loss as a function of carbon-oxygen core mass at each metallicity $Z$ as computed in @farmer:19. The crosses show the values from Tab. 1 @farmer:19.<span data-label="fig:fit_DM_PPI"></span>](fit_DM_PPI.pdf){width="0.75\linewidth"}

The top-down approach of , imposes $\Delta M_\mathrm{PPI}=0$ at the edge of the PPI regime, which produces a smooth BH mass distribution. We fit the metallicity-dependent naked helium core PPI simulations of @farmer:19 to obtain $\Delta M_\mathrm{PPI} \equiv \Delta M_\mathrm{PPI}(M_\mathrm{CO},Z)$ as a function of the carbon-oxygen core mass (). While the fit of @farmer:19 provides the remaining mass after PPI, this is only an estimate of the BH mass because of the the other mass loss processes that might occurr [e.g., @renzo:20csm; @powell:21; @rahman:22]. Our approach fits the mass removed by PPI only, which is what is directly computed in @farmer:19.

The dashed curves in each panel of show the fit for each metallicity computed in @farmer:19. We neglect the (weak) metallicity dependence of the minimum core mass for PPI, and we fit the publicly available data for initial He core masses between $38-60\,M_\odot$. We emphasize that @farmer:19 only simulated helium cores. In the case of a star with a H-rich envelope which is still present at the onset of the pulsations and if it is both extended and red, it can be easily removed by the first pulse [@woosley:17; @renzo:20csm]. Thus the H-rich mass of red supergiants should be added to the $\Delta M_\mathrm{PPI}$ provided here. It is unclear what occurs in cases when the envelope is compact and blue [e.g., @dicarlo:19; @renzo:20merger; @costa:21].

The amount of mass lost in PPI is sensitive to convection inside the star [@renzo:20conv] and the assumed nuclear reaction rates [@farmer:19; @farmer:20; @costa:21; @woosley:21; @mehta:21], which can introduce uncertainties up to $\sim{}20\%$ on the maximum BH mass. The accuracy of our fit is comparable to these uncertainties.

Acknowledgements {#acknowledgements .unnumbered}
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MR is grateful to R. Luger for help with showyourwork [@luger:21]. The code associated to this paper is publicly available at <https://github.com/mathren/top_down_compact_obj_mass> and the input data are loaded from <https://zenodo.org/record/3346593>. LvS acknowledges partial financial support from the National Science Foundation under Grant No. (NSF grant number 2009131), the Netherlands Organisation for Scientific Research (NWO) as part of the Vidi research program BinWaves with project number 639.042.728 and the European Union's Horizon 2020 research and innovation program from the European Research Council (ERC, Grant agreement No. 715063).

{'source': 'arXiv_out/2201'}