The star formation history in the solar neighborhood as told by massive white dwarfs

TL;DR

This paper uses Gaia data to analyze the luminosity function of massive white dwarfs in the solar neighborhood, reconstructing the star formation history over the past 8 billion years.

Contribution

It provides a novel reconstruction of the local star formation rate using white dwarf cooling ages and Gaia data, revealing multiple star formation episodes.

Findings

Star formation rate increased from zero in early Galaxy

Peak star formation occurred 6-7 Gyr ago

Recent star formation burst traces are tentative

Abstract

White dwarfs are the remnants of low and intermediate mass stars. Because of electron degeneracy, their evolution is just a simple gravothermal process of cooling. Recently, thanks to Gaia data, it has been possible to construct the luminosity function of massive (0.9 < M/Msun < 1.1) white dwarfs in the solar neighborhood (d < 100 pc). Since the lifetime of their progenitors is very short, the birth times of both, parents and daughters, are very close and allow to reconstruct the (effective) star formation rate. This rate started growing from zero during the early Galaxy and reached a maximum 6-7 Gyr ago. It declined and ~5 Gyr ago started to climb once more reaching a maximum 2 - 3 Gyr in the past and decreased since then. There are some traces of a recent star formation burst, but the method used here is not appropriate for recently born white dwarfs.

Tables

Table 1. Table 1: Total star formation rate Ψ Ψ \Psi  (M ⊙ Gyr -1 pc -3 ), age, t 𝑡 t and time interval Δ ​ t Δ 𝑡 \Delta t (Gyr) obtained from each luminosity function bin. Subindexes s and ns correspond to the cases with and without sedimentation.

$lo{g}_{10}\left(L/L_{\odot }\right)$	$t_s$	$\Delta t_s$	${\log }_{10}{\Psi }_s$	$t_{\mathrm{ns}}$	$\Delta t_{\mathrm{ns}}$	${\log }_{10}{\Psi }_{\mathrm{ns}}$
-1.20	0.05	0.04	-2.794	0.05	0.04	-2.794
-1.70	0.12	0.16	-2.553	0.12	0.16	-2.553
-2.30	0.41	0.43	-2.655	0.41	0.42	-2.643
-2.80	0.97	0.81	-2.780	0.91	0.64	-2.678
-3.10	1.80	0.70	-2.546	1.53	0.52	-2.418
-3.30	2.59	0.88	-2.468	2.13	0.67	-2.350
-3.50	3.53	0.99	-2.600	2.86	0.80	-2.508
-3.70	4.58	1.11	-2.747	3.75	0.98	-2.694
-3.90	5.75	1.23	-2.753	4.82	1.16	-2.728
-4.10	7.06	1.46	-2.667	6.09	1.43	-2.660
-4.30	8.88	2.58	-2.885	7.89	2.57	-2.884
-4.50	11.95	2.82	-3.403	10.96	2.82	-3.403
-4.70	14.13	1.66	-4.123	13.14	1.68	-4.130

Equations

$$N(l)={\int }_{M_{\mathrm{i}}}^{M_{\mathrm{s}}}\Phi (M)\Psi [T-t_{\mathrm{cool}}(l,M)-t_{\mathrm{PS}}(M)]{\tau }_{\mathrm{cool}}(l,M)dM$$

$$N(l)=\underset{0}{\overset{t_b^{up}}{\int }}K\left(l,t_b\right)\Psi \left(t_b\right)d{t}_b$$

$$K\left(l,t_b\right)=\Phi \left[M\left(t_b\right)\right]{\tau }_{cool}\left[l,M\left(t_b\right)\right]\frac{dM\left(t_b\right)}{d{t}_b}$$

$$N(l,T)=\frac{1}{\Delta l}\underset{t}{\int }\underset{M}{\int }\Phi \left(M\right)\Psi \left(t\right)dMdt$$

$$T-t_{cool}\left(M,l-0.5\Delta l\right)\le t+t_{MS}\left(M\right)\le T-t_{cool}\left(M,l+0.5\Delta l\right)$$

$$N(l,T)\simeq \frac{⟨\Psi ⟩}{\Delta l}\underset{\Delta M}{\int }\Phi \left(M\right)\Delta t_{cool}\left(l,M\right)dM$$

$$\Delta t_{cool}=t_{cool}\left(l+0.5\Delta l,M\right)-t_{cool}\left(l-0.5\Delta l,M\right)$$

$$⟨\Psi ⟩=\frac{N(l,T)\Delta l}{\underset{\Delta M}{\int }\Phi \left(M\right)\Delta t_{cool}\left(l,M\right)dM}$$

$$⟨t⟩=\frac{\underset{\Delta M}{\int }\Phi (M)tdM}{\underset{\Delta M}{\int }\Phi (M)dM}$$

$$⟨\Delta t⟩=\frac{\underset{\Delta M}{\int }\Phi (M)\Delta tdM}{\underset{\Delta M}{\int }\Phi (M)dM}$$

Full text

Galaxy:evolution-Galaxy: solar neighborhood-stars: white dwarfs

††journal: ApJ Letters

1 Introduction

The luminosity function is defined as the number of white dwarfs of a given luminosity per unit volume (or galactic disk surface unit, for instance) and magnitude interval (WDLF from now):

[TABLE]

where $T$ is the age of the population under study, $l=-\log(L/L_{\odot})$, $M$ is the mass of the parent star (for convenience all white dwarfs are labeled with the mass of the main sequence progenitor), $t_{\rm cool}$ is the cooling time down to luminosity $l$, $\tau_{\rm cool}=dt/dM_{\rm bol}$ is the characteristic cooling time, $M_{\rm s}$ is the maximum mass of a main sequence star able to produce a white dwarf, and $M_{\rm i}$ is the minimum mass of the main sequence stars able to produce a white dwarf of luminosity $l$, i.e. is the mass that satisfies the condition $T=t_{\rm cool}(l,M)+t_{\rm PS}(M)$ and $t_{\rm PS}$ is the lifetime of the progenitor star. The remaining quantities, the initial mass function (IMF, from now), $\Phi(M)$, and the star formation rate (SFR, from now), $\Psi(t)$, are not known a priori and depend on the astronomical properties of the stellar population under study. Since the total density of white dwarfs of a given population is usually not well known, it is customary to normalize the computed luminosity function to a bin with a small error bar in order to compare theoretical and observational data. For instance, in the case of the disk this bin is usually $l=3$. Therefore, if the observed luminosity function and the evolutionary behavior of white dwarfs are known it is possible to obtain information about the properties of the population under study. Evidently, given the nature of the problem, there is always a degeneracy between the galactic properties (SFR and IMF) and the adopted stellar models.

The process of obtaining such information can be formulated as follows. Let be $t_{\rm b}=T_{disk}-t_{cool}(l,M)-t_{PS}(M)$ the time at which the progenitor of the white dwarf was born and $M=M(t_{\rm b})$ the mass of the star that, being born at this time, is able to produce a white dwarf of luminosity $l$ at present. Equation 1 can be written as:

[TABLE]

with

[TABLE]

The kernel, $K(l,t_{\rm b})$, of this integral function is not symmetric in $l$ and $t_{\rm b}$ and it has a quite complicated behavior. Consequently, according the Picard-Lindelöf’s theorem, $\Psi$ cannot be directly obtained and the unicity of the solution is not guaranteed (Isern et al. 1995).

One way to tackle the problem is to optimize the parameters of some trial functions comparing, after defining some weight function, models with data (Isern et al. 1999). Obviously, this solution is optimal within the context of the adopted model, which might not correspond with the reality. Another way consists on, starting from a simple initial guess of the SFR, iteratively improve the solution using all the observational bins until a satisfactory solution is found (Rowell 2013a). This solution is quite sensitive to the adopted metallicity and IMF, but not to the DA non-DA white dwarf ratio nor the relationship between the mass of the white dwarf and that of the progenitor. All in all, the quality of the final solution essentially depends on the quality of the observational data.

Finally, if the luminosity function is restricted to massive white dwarfs the SFR can be directly obtained (Diaz-Pinto et al. 1994). This method, however, has suffered from the scarcity of high mass white dwarfs known. In an early work, this SFR was obtained from the data of Sion et al. (1988) and Bergeron, Saffer & Liebert (1992), and from Legget et al. (1998) respectively, but the relatively small number of stars in the sample prevented to obtain firm conclusions (Isern et al. 1999). Fortunately this situation has recently changed thanks to the work of Tremblay et al. (2019) who have been able to build a reliable and precise luminosity function of massive stars using the data provided by Gaia.

2 Massive white dwarfs and the star formation rate

This luminosity function, averaged over an interval of luminosity $\Delta l$, can also be directly computed as follows (Isern et al. 1999). Assume a stellar population that forms at a rate $\Psi(t)$. After a time $T$, the number of white dwarfs that have a luminosity $l$ per unit of luminosity interval is given by

[TABLE]

where, as before, $M$ is the mass of the parent star, and the integral is constrained to the domain

[TABLE]

for all the stars able to produce a white dwarf.

If the integral is restricted to massive white dwarfs, i.e. those for which it is possible to neglect the lifetime of the progenitor in front of the cooling time, and $\Psi(t)$ is smooth enough111This method is also valid for white dwarfs with masses within a limited enough range of values., then

[TABLE]

with

[TABLE]

and consequently,

[TABLE]

[TABLE]

[TABLE]

It is important to notice here that the star formation rate obtained in this way is an effective one in the sense that it recovers the present age distribution of the sample, but does not take into account the secular evolution of the sample mainly due to radial migrations and height inflation. On another hand, hidden WD in binaries and non-resolved double degenerates can bias the sample, and double degenerate mergers can reduce the density of WD in some bins and, in the case they do not explode as SNIa reappear as newly born hot single WD with the corresponding density increase of younger bins, thus modifying the SFR deduced from these data. The importance of this effect is small given the present level of precision, but it will be necessary to include it in order to interpret future high precision WDLFs.

3 Results and conclusions

Table 1 shows the values taken by $\left\langle\Psi\right\rangle$, $\left\langle t\right\rangle$ and $\left\langle{\Delta t}\right\rangle$ using the Tremblay et al. (2019) data and the BaSTI models222 Cooling models publically available at: http://albione.oa-teramo.inaf.it. for DA white dwarfs. Models labeled ns take only into account the release of latent heat upon crystallization, while models labeled s take also into account the gravitational energy released by the sedimentation induced by the changes of solubility during the crystallization process. Both families of models are built with the chemical profiles predicted by the evolution of the progenitor which depend on the mass (Salaris et al. 2010). The relationship between the masses of the progenitor and white dwarf is that found by El-Badry et al. (2018)333 The results obtained with the Catalan et al. (2008) initial final mass relationship are similar. , while the IMF is that of Salpeter truncated at 0.1 M⊙ and normalized to the unit mass.