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Table 2.

Evolution of a zero-age main-sequence binary towards FN Sgr.

Time M1 M2 Type1 Type2 Porb Event
(Myr) (M) (M) (days)
0.0000 2.200 1.360 MS MS 2000.000 Zero-age MS binary
840.5808 2.199 1.360 SG MS 2001.400 Change in primary type
853.4929 2.199 1.360 FGB MS 2001.459 Change in primary type
868.2086 2.198 1.360 CHeB MS 2002.091 Change in primary type
1139.2842 2.196 1.360 E-AGB MS 2004.652 Change in primary type
1156.3479 2.196 1.360 TP-AGB MS 1997.691 Change in primary type
1158.3386 2.116 1.360 TP-AGB MS 1785.599 Onset of CE evolution (αCE = 0.3 and αint ≈ 0.82)
1158.3386 0.606 1.360 WD MS 630.000 End of CE evolution
3096.4561 0.606 1.360 WD SG 630.259 Change in secondary type
3349.8662 0.606 1.360 WD SG 630.325 WD starts to crystallize
3349.8662 0.606 1.360 WD SG 630.325 WD weak magnetic field is generated deep in the core
3624.6724 0.606 1.360 WD FGB 630.401 Change in secondary type
3949.8662 0.606 1.359 WD FGB 630.713 WD magnetic field penetrates the surface
4041.4845 0.606 1.335 WD FGB 578.405 WD is spun up (accreted enough angular momentum)
4041.4845 0.606 1.335 WD FGB 578.405 WD develops a super-equipartition strong magnetic field
4041.7263 0.606 1.332 WD FGB 568.701 Onset of SySt phase (RL filling factor ≈0.90)
4041.7403 0.606 1.331 WD FGB 567.310 Binary looks like FN Sgr
4041.7882 0.611 1.220 WD FGB 497.086 Onset of CE evolution (αCE = 0.3 and αint = 1.0)
4041.7882 0.611 0.452 WD WD 0.954 End of CE evolution

Notes. For the pre-CE and post-CE evolution, we used the MESA code with the assumptions described in Sect. 3. For the CE evolution, we computed the post-CE orbital period using Eqs. (1) and (2), assuming αCE = 0.3 and αCE ≈ 0.82. The terms M1 and M2 and Type1 and Type2 are the masses and stellar types of the primary and secondary, respectively. Porb is the orbital period and the last column corresponds to the event occurring to the binary at the given time in the first column. The row in which the binary has the present-day properties of FN Sgr is highlighted in boldface. For the WD evolution, we interpolated the evolutionary sequences calculated by Bédard et al. (2020) to estimate the mass fraction of the crystallized matter. We assumed a magnetic field diffusion timescale of ∼600 Myr and that the amount of angular momentum accreted by the WD before the binary became a SySt was enough to spin up the WD towards the super-equipartition regime. Abbreviations: MS (main sequence star), SG (subgiant star), FGB (first giant branch star), CHeB (core helium burning), E-AGB (early asymptotic giant branch star), TP-AGB (thermally-pulsing asymptotic giant branch star), WD (white dwarf), RLOF (Roche lobe overflow), CE (common envelope), SySt (symbiotic star).

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