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Unified gravity and Rossby mode analysis in accreting white dwarf stars

Kumar, Praphull; Townsley, Dean

Abstract

Dwarf novae are a subset of cataclysmic variables that accrete material intermittently in short-duration outbursts with sometimes long quiescent intervals in between. During the quiescent state, the white dwarf (WD) photosphere may be observable. Some of these systems show periodic variability consistent with a non-radial oscillations. Asteroseismology has become a unique tool for the measurement of internal structure of the WDs, such as their masses, radii, temperatures and rotation profiles. A few stable periodicities have been observed for these systems, but the lack of complete and accurate theoretical models has hindered the real diagnosis of the observed pulsations. Though the associated pulsations in accreting WDs are thought to be g-modes, some work in the literature suggests that these pulsations could be Rossby modes (r-modes). Here, to elucidate this, we present a first simultaneous analysis of g- and r-mode pulsations in accreting WDs including a full computation of visibility accounting for the distribution of variation over the WD surface. We show that, up to the second lowest degree (l=2), neither g nor r-modes have a clear advantage in visibility. Although, a few retrograde r-mode orders exhibit a larger visibility, the low-order g modes possess higher frequency in the star's frame making them more likely to be driven within the convective driving scenario commonly applied to isolated WDs. Therefore, we favor a g-mode origin for the observed periods in accreting WDs. We also explore how the normal mode frequencies vary with changes in properties of the WD such as mass, core temperature and accreted layer thickness. We propose a potential new method to identify the mode orders based on time variation of frequencies during the months of cooling after the accretion event.

Full text

Unified gravity and Rossby mode analysis in accreting white dwarfs Praphull Kumar PhD Candidate, The University of Alabama, USA Supervisor: Dr. Dean Townsley •Cataclysmic variable star (CV) -- a compact binary system (orbital period ≤ 1 day) – WD [primary] + low-mass main sequence star [secondary]. •Mass transfer from MS star to WD via Roche-lobe overflow. •CV WDs rotate rapidly, spun up due to accretion •Observed masses: 0.8 – 1.2 𝑀⊙ •Accreting WDs in CV pass through several accretion outbursts. •Some of the CV WDs exhibit pulsations behavior. •Seismology has become an advanced tool to probe these systems. White Dwarfs in CVs Szkody et al. 2016 But we don’t know the mode orders!! Propagation Diagram •g-modes propagate when 𝜔 < 𝑁%and 𝜔 < 𝑆" (restored by buoyancy) •p-modes propagate when 𝜔 > 𝑁%and 𝜔 > 𝑆" (restored by pressure) •r-mode (restored by Coriolis force) Kumar & Townsley 2023 Kumar & Townsley, 2023 g-modes with star’s spin 𝜔#= 𝜔$+mΩ Do we need to consider other modes, too? Rossby modes g/r-mode branching LTE à In Non-rotating limit: λ = ℓ((ℓ + 1) •The solutions to the eigenvalue problem in the non-rotating case is 𝑌 ,-. •In the rotating case à Hough functions •Spin parameter, q = ./ 0 m > 0: retrograde (sign convention change) 𝐿!(Θ) = −𝜆Θ( Kumar & Townsley 2025, submitted 0.78𝑀⊙ Even k = 0, 2 : even modes Odd k = -3, -1, 1, 3 : odd modes Solutions to LTE: Hough Functions R-mode: q = 2.68, n = 6 Kumar & Townsley 2025, submitted 𝜇 = 0: Equator Pole 2-D Eigenfuntions Solid core ~ 35% Visibility Calculation Normalized Hough functions Eulerian Pressure perturbation at surface Line of sight transformation Limb-darkening ~ 60% Summary and Future Work! •We were able to constrain both the gand r-modes in the accreting white dwarf system. •Although a few r-mode show stronger visibilities than the g-mode, they are unlikely to be driven and thus be part of observables. •Even with a solid core, the TAR fails near the central part of the WD. •Full 2D problem: Solutions are no longer separable from 2D to 1D. •Both gand r-modes: mixed mode character – the same mode will have an inertial behavior near the center and the gravity mode near the surface. •A unified solution is underway – beyond TAR. •A new method of mode identification in accreting system using 𝑃;<= vs 𝑃. Thanks! Questions and Comments? Mode Identification •Rate of change of the mode frequency from three to six months since the DN outburst •Filled symbols: mode driven w.r.t. the star’s frame •Open symbols: Unlikely to be driven and be observed •The minimum driving period: 192 seconds. We choose this with what is available from the observations. Extra Kumar & Townsley 2025, in Prep. g-mode visibility Extra 0.0 0.2 0.4 0.6 0.8 1.0 Visibility/pK.E. Inc.=11± (m, k) = (1, -1)[r] (m, k) = (1, -2)[r] (m, k) = (1, -3)[r] (m, k) = (2, -1)[r] (m, k) = (2, -2)[r] Driving frequency Inc.=11± GW Lib Obs: 236 s GW Lib Obs: 377 s GW Lib Obs: 646 s 0.0 0.2 0.4 0.6 0.8 1.0 Visibility/pK.E. Inc.=45±Inc.=45± 0 1000 2000 3000 4000 5000 6000 Frequency (corot) in µHz 0.0 0.2 0.4 0.6 0.8 1.0 Visibility/pK.E. Inc.=70± 0 1000 2000 3000 4000 5000 6000 Frequency (inertial) in µHz Inc.=70± r-mode visibility Extra Red: + pressure perturbation Blue: - pressure perturbation How would surface look like? Kumar & Townsley 2025, submitted Prograde g-modeq = 1.57, n = 6 q = 3.04, n = 23 Standard final equations l(l+1) – slow rotation otherwise solve for ƛ 𝐿!Θ = −𝜆Θ( 𝝁 = 𝒄𝒐𝒔𝜽 % N >> 2Ω and N>> ⍵ Neglecting horizontal component Ω𝑠𝑖𝑛θ Extra TAR separated obtained solutions >> r and 𝜽 and then put it back together to obtain the final Solutions. rotating case Extra