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With the precision now afforded by modern space-based photometric observations from the retired K2 and current TESS missions, the effects of general relativity (GR) may be detectable in the light curves of pulsating white dwarfs (WDs). Almost all WD models are calculated using a Newtonian description of gravity and hydrodynamics. To determine if the inclusion of GR leads to observable effects, we used idealized models of compact stars and made side-by-side comparisons of mode periods computed using a: (i) Newtonian and (ii) GR description of the equilibrium structure and nonradial pulsations. For application to WDs, it is only necessary to include the first post- Newtonian (1PN) approximation to GR. The mathematical nature of the linear nonradial pulsation problem is then qualitatively unchanged and the GR corrections can be written as extensions of the classic Dziembowski equations. As such, GR effects might easily be included in existing asteroseismology codes. The idealized stellar models are (i) 1PN relativistic polytropes and (ii) stars with a cold degenerate electron equation of state featuring a near-surface chemical transition from μe = 2 to μe = 1, simulating a surface hydrogen layer. A comparison of Newtonian and 1PN normal mode periods reveals fractional differences in the order of the surface gravitational redshift z. For a typical WD, this fractional difference is ∼10−4 and is greater than the period uncertainty σΠ/Π of many WD pulsation modes observed by TESS. Consistent theoretical modeling of periods observed in these stars should, in principle, include GR effects to 1PN order. 

ABSTRACT We report the discovery of two apparently isolated stellar remnants that exhibit rotationally modulated magnetic Balmer emission, adding to the emerging DAHe class of white dwarf stars. While the previously discovered members of this class show Zeeman-split triplet emission features corresponding to single magnetic field strengths, these two new objects exhibit significant fluctuations in their apparent magnetic field strengths with variability phase. The Zeeman-split hydrogen emission lines in LP 705−64 broaden from 9.4 to 22.2 MG over an apparent spin period of 72.629 min. Similarly, WD J143019.29−562358.33 varies from 5.8  to 8.9 MG over its apparent 86.394 min rotation period. This brings the DAHe class of white dwarfs to at least five objects, all with effective temperatures within 500 K of 8000 K and masses ranging from $$0.65\,\,{\text{to}}\,\,0.83\, {\rm M}_{\odot }$$. 

Abstract PG 1159-035 is the prototype of the PG 1159 hot (pre-)white dwarf pulsators. This important object was observed during the Kepler satellite K2 mission for 69 days in 59 s cadence mode and by the TESS satellite for 25 days in 20 s cadence mode. We present a detailed asteroseismic analysis of those data. We identify a total of 107 frequencies representing 32ℓ= 1 modes, 27 frequencies representing 12ℓ= 2 modes, and eight combination frequencies. The combination frequencies and the modes with very highkvalues represent new detections. The multiplet structure reveals an average splitting of 4.0 ± 0.4μHz forℓ= 1 and 6.8 ± 0.2μHz forℓ= 2, indicating a rotation period of 1.4 ± 0.1 days in the region of period formation. In the Fourier transform of the light curve, we find a significant peak at 8.904 ± 0.003μHz suggesting a surface rotation period of 1.299 ± 0.002 days. We also present evidence that the observed periods change on timescales shorter than those predicted by current evolutionary models. Our asteroseismic analysis finds an average period spacing forℓ= 1 of 21.28 ± 0.02 s. Theℓ= 2 modes have a mean spacing of 12.97 ± 0.4 s. We performed a detailed asteroseismic fit by comparing the observed periods with those of evolutionary models. The best-fit model hasT_eff= 129, 600 ± 11 100 K,M_*= 0.565 ± 0.024M_⊙, and $\log g={7.41}_{-0.54}^{+0.38}$, within the uncertainties of the spectroscopic determinations. We argue for future improvements in the current models, e.g., on the overshooting in the He-burning stage, as the best-fit model does not predict excitation for all of the pulsations detected in PG 1159-035.