The secondary maximum of T CrB caused by irradiation of the red giant by a cooling white dwarf

Both the 1866 and 1946 outbursts of the recurrent symbiotic nova T CrB have displayed a mysterious secondary maximum peaking in brightness ~5 months past the primary one. Common to all previous modeling attempts was the rejection of plain irradiation of the red giant (RG), on the basis that the secondary maximum of T CrB would have been out of phase with the transit at superior conjunction of the RG. Implicit to this line of reasoning is the assumption of a constant temperature for the white dwarf (WD) irradiating the red giant. I show by radiative modeling that irradiation of the RG by a cooling WD nicely reproduces the photometric evolution of the secondary maximum, both in terms of brightness and color, removes the phasing offset, and provides a straightforward explanation that will be easy to test at the next and imminent outburst.


INTRODUCTION
T CrB belongs to the rare group of symbiotic recurrent novae, of which only ∼four are known in the Galaxy, the most famous being RS Oph.The last eruption of T CrB occoured in 1946 and its light curve was a near-perfect replica of the previous outburst in 1866 (Pettit 1946a).The lightcurve of both events (separated by exactly 128 orbital revolutions) is characterized by the presence of a broad secondary maximum (II-Max hereafter), which is seen only in T CrB.
Different explanations (including an accretion episode, irradiation of a tilted disk, and a second and separate nova eruption) have been proposed for II-Max (eg.Webbink 1976;Hachisu & Kato 1999;Schaefer 2023).Common to all of them is the rejection of a plain irradiation of the red giant (RG), on the basis that II-Max is out of phase with the transit at superior conjunction of RG (ψ=0 in Figure 1).Implicit to this line of reasoning is the assumption of a constant temperature for the WD irradiating the red giant.
Supported by the results of detailed radiative modeling, I will show that irradiation of the RG by a cooling WD nicely fits the photometric observations of II-Max, both in terms of brightness and color, removes the phasing problem, and provides a straightforward explanation that will be easy to test at the next, imminent outburst (Munari et al. 2016;Schaefer 2023;Munari 2023).

RADIATIVE MODELING
A radiative modeling for II-Max has been carried out in physical units, placing T CrB at the 916 pc distance derived by Gaia and adopting a E B−V =0.05 reddening, a mass of 1.3 M ⊙ for the white dwarf (WD) and 0.93 M ⊙ for the red giant (RG), 227.56 days as the orbital period, i=65 • for the orbital inclination, and a null eccentricity (Fekel et al. 2000).The WD is taken to radiate isotropically as a blackbody, while the surface of the Roche-lobe filling RG is divided into a 256×256 mesh grid, with each area bin radiating according to model atmospheres taken from Castelli & Kurucz (2003), and interpolated to local T eff and log g.Coefficients for linear gravity darkening are derived from Claret & Bloemen (2011).A fraction η of the radiation arriving from the WD on the RG is locally absorbed and re-thermalized, the remaining 1−η is scattered out as it is.A T eff =3500 • K is adopted for the shadowed regions of RG, in line with its M3III spectral classification.The binary system is followed through orbital revolution, and at each step the emitted spectrum is integrated through the profile of Landolt B,V bands and magnitudes computed, with flux zero-points taken from Bessell et al. (1998).The lightcurve of the 1946 outburst of T CrB is presented in Figure 1 (dots).It is built from Pettit (1946a) observations ported to modern Landolt V -band by comparing to APASS survey (Henden & Munari 2014) the quoted magnitudes for the eight original comparison stars.The spectral evolution of T CrB during the 1946 outburst has been described in detail by eg.Sanford (1946Sanford ( , 1947Sanford ( , 1949) ) and Morgan & Deutsch (1947), indicating how the WD was very hot when II-Max begun on day +109, with coronal lines of [FeX] and [FeXIV] being persistently strong since day +4, while [FeVII] was still absent.At the end of II-Max the WD was still rather hot, albeit cooler, with spectra showing [FeX] and [FeVII], but no more [FeXIV].Also RS Oph displayed for long a very hot WD during the 2006 and 2021 outbursts, till at least day +86 as proved by the prominence of [FeX], [FeXIV] in optical spectra (Munari & Valisa 2022) and the strong super-soft emission in X-rays observations (Page et al. 2022).
During the radiative modeling runs only the WD and the irradiated RG were considered, the contribution from the nova ejecta to overall brightness being irrelevant: in fact, the ejecta were already optically thin by day +4 (coronal lines prominent), the M3III spectrum of the RG returned visible in the blue by day +13, and the classical nebular emission lines ever developed.No accretion disk is considered either, by analogy with RS Oph in which the disk begins reforming only ∼120 days past disappearance of coronal lines (Munari & Tabacco 2022).The temperature of the WD at the beginning of II-Max is assumed to be 220,000 K consistent with a photoionization origin for [FeXIV], and the radius set to 0.23 R ⊙ to fit the constant brightness exhibited by T CrB during the weeks preceding II-Max.While the temperature of the WD was let to change, its radius has been kept fixed through II-Max.

Figure 1 .
Figure 1.The secondary maximum of T CrB is accurately reproduced (orange line) by irradiating the red giant with the white dwarf companion cooling according to the temperature profile plotted in the top panel.The dashed line shows for reference the ellipsoidal modulation computed for the Roche-lobe filling red giant.The inset present the evolution of (B−V ) color.Days are counted from primary maximum on JD 2431860.854(= 1946 Feb 9.354 UT), and phase ψ is reckoned from RG passage at superior conjunction (ψ=0.0 on day +187 and JD 2432048).At the bottom, examples of the computed radiative models (the crosses highlight the position of the WD).