1, J.J. Hermes2,3, B. H. Dunlap2 Bounded by the Base of...
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Insights from Pulsation Modes Bounded by the Base of the Convection Zone M. H. Montgomery 1 , J.J. Hermes 2,3 , B. H. Dunlap 2 , D. E. Winget 1 , Keaton J. Bell 1 , Judith Provencal 4 , J. C. Clemens 2 , Stephen Fanale 2 1 University of Texas at Austin, 2 University of North Carolina at Chapel Hill, 3 Hubble Fellow, 4 University of Delaware By allowing the ML2/α parameter to vary with T eff we can match the observed transition where modes appear broadened Long-period pulsations are eventually bounded by the base of the convection zone Observed correlation of mode width and convection zone depth Fontaine & Brassard 2008, PASP, 120, 1043 Montgomery, Provencal, Kanaan, et al. 2010, ApJ, 716, 84 Fanale, Hermes, Clemens, et al. 2016, in preparation Provencal, Montgomery, Shipman, et al. 2015, ASPC, 493, 187 Tremblay, Bergeron & Gianninas, 2011, ApJ, 730, 128 Tremblay, Ludwig, Freytag, et al. 2015, ApJ, 799, 142 References: In model atmospheres, the depth of the convection zone depends on T eff , log(g), and the convective efficiency, which is parameterized by α, the ratio of the mixing length to the pressure scale height. Because it moves the base of the convection zone in the models, different choices of α result in different predictions for P cross . The solid lines in the figure at right show this effect and also show that, for constant α, P cross moves to shorter periods as the convection zone deepens with cooler temperatures. Fanale et al. find a notable dividing line between wide and narrow modes at 800 s. Furthermore, they note that modes with periods > 800 s are found in stars with of T eff < 11,500 K. An explanation for this effect is that P cross ≈ 800 s for models with T eff < 11,500 K. In the figure at right we plot P cross versus T eff (assuming log(g) = 8.0) for different parameterizations of convection: ML2/α = 0.6, 0.8, 1.0, 1.6, as well as a functional form that depends on temperature, α(T). Here α LC (T) denotes the effective value of α LC (T) as derived from convective light curve fits to the nonlinear pulse shape (see below). We include a line showing 1.1 x α LC (T), and one showing the 3D convection simulations of Tremblay et al. (2015) as α PT (T). DAV white dwarfs pulsate in nonradial g-modes. The propagation region of these modes is defined as the region where the mode frequency is less than both the B-V (“buoyancy”) and the Lamb (“acoustic”) frequencies. Short-period/high-frequency g-modes have an outer turning point defined by the Lamb frequency, whereas long-period/low-frequency modes have an outer turning point defined by the base of the convection zone (we denote the dividing period as P cross ; see plot at left). As DAVs pulsate, the local surface temperature (T eff ) changes. The convection zone responds quasi-statically to the pulsations, becoming deeper as the local temperature decreases and shallower as it increases. Modes whose outer turning point is defined by the Lamb frequency will be mostly unaffected by these temperature changes, whereas modes whose outer turning point is the base of the convection zone will be more perturbed; the changing size of the g-mode cavity for these modes could lead to a lack of phase stability. We have recently found that DAVs observed with the Kepler spacecraft show two distinct kinds of modes: those with a coherent phase over several months and those lacking coherence. We identify these as modes on either side of P cross . A Fourier transform reveals these as modes with power in a single frequency bin and modes with power spread across a few adjacent bins (Hermes et al., private communication). The dotted horizontal lines show the propagation region for all 16 of the observed modes in one of the stars presented in Fanale et al. (with T eff =10,990 K, 0.55 M ⦿ ): we mark in blue modes whose Fourier peaks have a HWHM < 0.3 μHz and red for those with HWHM > 0.3 μHz. The gray solid line, which shows a mode with period P=P cross that barely reaches the base of the convection zone, neatly divides the set into “wide” (red) and “narrow” (blue) modes. The single exception (red dotted line above the gray line) is naturally explainable as an l=2 mode. We note that any fixed value of the ML2/α appears to predict P cross too short for the many DAVs with T eff < 11,000 K, but that the 1.1 x α LC (T) curve predicts P cross ≈ 800 s across a wide range of temperatures; a more detailed analysis will be required to evaluate the relative merit of the different relations. This analysis can provide an alternate method for constraining the convective efficiency in DAV stars. We show the convective response timescale τ as a function of T eff as derived from convective light curve fits, following the methods Montgomery et al. (2010) and Provencal et al. (2015) with an updated sample of DAVs. These T eff include the 3D dependence of convection using the corrections from Tremblay et al. (2013). The curves show the predictions of ML2 for different values of α at the base of the convection zone. We note that α < 0.8 is favored for T eff ≈ 11,500 K. The nonlinear pulse shape fitting predicts that α varies by of order a factor of two through the instability strip, from roughly 1.6 – 0.6. Recent 3D simulations of convection by Tremblay et al. (2015) estimate a similar change in ML2/α at the Schwarzschild boundary, ranging from 1.20-0.68 for 13,000 < T eff < 10,500 K. In the Figure at right we show the half width at half maximum (HWHM) of Lorentzian functions fit to all significant peaks in the power spectra of the 21 DAVs observed so far by the Kepler space telescope. We see that virtually all modes with HWHM > 0.3 μHz have periods longer than 800 s. In addition, assuming ML2/α =0.7 convection and l=1 pulsation modes, we have computed the depth of the convection zone for the observed T eff and log g values for each star; modes whose periods are long enough that they propagate to the base of their star’s convection zone are colored red, while modes with shorter periods are colored blue. We see that there is a strong correlation between mode width and whether the mode is expected to reflect off the base of the convection zone. For ML2 convection, we have varied the α parameter from 0.4 to 1.1; we find that α ≈ 0.7 does the best job of dividing the data into long period/large HWHM and short period/small HWHM. We note that the convection zone depth derived from 3D convection (Tremblay et al. 2015) provides an equally good division, as does that derived from convective light curve fits. We emphasize that these data provide important observational constraints on the convection zone depth of ZZ Ceti stars. In this era of precision asteroseismology, this has consequences not only for the driving of pulsation modes, but also for the computed periods of long period l=1 and 2 modes (Fontaine & Brassard 2008).
Transcript of 1, J.J. Hermes2,3, B. H. Dunlap2 Bounded by the Base of...
Insights from Pulsation Modes Bounded by the Base of the
Convection Zone
M. H. Montgomery1, J.J. Hermes2,3, B. H. Dunlap2,D. E. Winget1, Keaton J. Bell1, Judith Provencal4,
J. C. Clemens2, Stephen Fanale21University of Texas at Austin, 2University of North Carolina at Chapel Hill,
3Hubble Fellow, 4University of Delaware
By allowing the ML2/α parameter to vary with Teff we can match the observed transition where modes appear broadened
Long-period pulsations are eventually bounded by the base of the convection zone
Observed correlation of mode width and convection zone depth
Fontaine & Brassard 2008, PASP, 120, 1043Montgomery, Provencal, Kanaan, et al. 2010, ApJ, 716, 84
Fanale, Hermes, Clemens, et al. 2016, in preparationProvencal, Montgomery, Shipman, et al. 2015, ASPC, 493, 187
Tremblay, Bergeron & Gianninas, 2011, ApJ, 730, 128Tremblay, Ludwig, Freytag, et al. 2015, ApJ, 799, 142References:
In model atmospheres, the depth of the convection zone depends on Teff, log(g), and the convective efficiency, which is parameterized by α, the ratio of the mixing length to the pressure scale height. Because it moves the base of the convection zone in the models, different choices of α result in different predictions for Pcross. The solid lines in the figure at right show this effect and also show that, for constant α, Pcross moves to shorter periods as the convection zone deepens with cooler temperatures. Fanale et al. find a notable dividing line between wide and narrow modes at 800 s. Furthermore, they note that modes with periods > 800 s are found in stars with of Teff < 11,500 K. An explanation for this effect is that Pcross ≈ 800 s for models with Teff < 11,500 K.
In the figure at right we plot Pcross versus Teff (assuming log(g) = 8.0) for different parameterizations of convection: ML2/α = 0.6, 0.8, 1.0, 1.6, as well as a functional form that depends on temperature, α(T). Here αLC (T) denotes the effective value of αLC(T) as derived from convective light curve fits to the nonlinear pulse shape (see below). We include a line showing 1.1 x αLC (T), and one showing the 3D convection simulations of Tremblay et al. (2015) as αPT(T).
DAV white dwarfs pulsate in nonradial g-modes. The propagation region of these modes is defined as the region where the mode frequency is less than both the B-V (“buoyancy”) and the Lamb (“acoustic”) frequencies. Short-period/high-frequency g-modes have an outer turning point defined by the Lamb frequency, whereas long-period/low-frequency modes have an outer turning point defined by the base of the convection zone (we denote the dividing period as Pcross; see plot at left).
As DAVs pulsate, the local surface temperature (Teff) changes. The convection zone responds quasi-statically to the pulsations, becoming deeper as the local temperature decreases and shallower as it increases. Modes whose outer turning point is defined by the Lamb frequency will be mostly unaffected by these temperature changes, whereas modes whose outer turning point is the base of the convection zone will be more perturbed; the changing size of the g-mode cavity for these modes could lead to a lack of phase stability.
We have recently found that DAVs observed with the Kepler spacecraft show two distinct kinds of modes: those with a coherent phase over several months and those lacking coherence. We identify these as modes on either side of Pcross. A Fourier transform reveals these as modes with power in a single frequency bin and modes with power spread across a few adjacent bins (Hermes et al., private communication). The dotted horizontal lines show the propagation region for all 16 of the observed modes in one of the stars presented in Fanale et al. (with Teff=10,990 K, 0.55 M⦿): we mark in blue modes whose Fourier peaks have a HWHM < 0.3 µHz and red for those with HWHM > 0.3 µHz. The gray solid line, which shows a mode with period P=Pcross that barely reaches the base of the convection zone, neatly divides the set into “wide” (red) and “narrow” (blue) modes. The single exception (red dotted line above the gray line) is naturally explainable as an l=2 mode.
We note that any fixed value of the ML2/α appears to predict Pcross too short for the many DAVs with Teff < 11,000 K, but that the 1.1 x αLC (T) curve predicts Pcross ≈ 800 s across a wide range of temperatures; a more detailed analysis will be required to evaluate the relative merit of the different relations. This analysis can provide an alternate method for constraining the convective efficiency in DAV stars.
We show the convective response timescale τ as a function of Teff as derived from convective light curve fits, following the methods Montgomery et al. (2010) and Provencal et al. (2015) with an updated sample of DAVs. These Teff include the 3D dependence of convection using the corrections from Tremblay et al. (2013). The curves show the predictions of ML2 for different values of α at the base of the convection zone. We note that α < 0.8 is favored for Teff ≈ 11,500 K.
The nonlinear pulse shape fitting predicts that α varies by of order a factor of two through the instability strip, from roughly 1.6 – 0.6. Recent 3D simulations of convection by Tremblay et al. (2015) estimate a similar change in ML2/α at the Schwarzschild boundary, ranging from 1.20-0.68 for 13,000 < Teff < 10,500 K.
In the Figure at right we show the half width at half maximum (HWHM) of Lorentzian functions fit to all significant peaks in the power spectra of the 21 DAVs observed so far by the Kepler space telescope. We see that virtually all modes with HWHM > 0.3 µHz have periods longer than 800 s. In addition, assuming ML2/α =0.7 convection and l=1 pulsation modes, we have computed the depth of the convection zone for the observed Teff and log g values for each star; modes whose periods are long enough that they propagate to the base of their star’s convection zone are colored red, while modes with shorter periods are colored blue. We see that there is a strong correlation between mode width and whether the mode is expected to reflect off the base of the convection zone.
For ML2 convection, we have varied the α parameter from 0.4 to 1.1; we find that α ≈ 0.7 does the best job of dividing the data into long period/large HWHM and short period/small HWHM. We note that the convection zone depth derived from 3D convection (Tremblay et al. 2015) provides an equally good division, as does that derived from convective light curve fits.
We emphasize that these data provide important observational constraints on the convection zone depth of ZZ Ceti stars. In this era of precision asteroseismology, this has consequences not only for the driving of pulsation modes, but also for the computed periods of long period l=1 and 2 modes (Fontaine & Brassard 2008).
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