arXiv subjects

Umin Lee

Publications and source records attributed to Umin Lee.

At least 19 recordsLinked to original sources

arXiv2025

Axisymmetric second order perturbations of rotating main sequence stars

We calculate the second order perturbations driven by oscillation modes of rotating stars. Assuming that the typical amplitude $a$ of oscillation modes is small, we expand the perturbed quantities as $Q=Q^{(0)}+Q^{(1)}+Q^{(2)}+\cdots$ where $Q^{(0)}$ represents the equilibrium state and $Q^{(1)}$ and $Q^{(2)}$ are the first order and second order perturbations in $a$, respectively. We assume that the first order perturbations are given by non-axisymmetric modes and the second order perturbations are axisymmetric. For the second order perturbations, we derive a set of linear partial differential equations, which have inhomogeneous terms due to the first order perturbations. For low frequency $g$- and $r$-modes and overstable convective (OsC) modes of main sequence stars, we calculate the second order velocity field $\mathbf{v}^{(2)}$ and find that prograde $g$-modes and OsC modes tend to accelerate and retrograde $r$-modes to decelerate $v_\phi^{(2)}$ in the surface equatorial regions where $v_\phi^{(2)}$ is the $\phi$ component of $\mathbf{v}^{(2)}$. Using the angular momentum conservation equation derived for waves, we discuss that low frequency $g$- and $r$-modes transport angular momentum between the inner and outer parts of the envelope. For OsC modes in the core resonantly coupled with envelope prograde $g$-modes, we find that they can transport angular momentum from the core to the outer envelope so that they tend to brake the core rotation. We also suggest that the OsC modes provide the outer envelope of rotating stars with the torque enough to support a decretion disc.

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arXiv2022

Non-linear excitation of low frequency modes by overstable convective modes in rotating stars

We discuss non-linear excitation and amplitude saturation of $g$-modes, $r$-modes and overstable convective (OsC) modes in early type main sequence stars, taking account of the effects of three-mode couplings on amplitude evolutions. OsC modes are rotationally stabilized convective modes in the convective core and they resonantly excite low frequency $g$-modes to obtain large amplitudes in the envelope when the rotation rate of the core is larger than critical rates. We use, for a network of three-mode couplings, amplitude equations governing the time evolution of the mode amplitudes where each of three-mode couplings is assumed to occur between two stable modes and one unstable mode. Assuming that the unstable modes in the couplings are OsC modes in the core and the stable modes are $g$- and $r$-modes in the envelope, we integrate the amplitude equations to see how the $g$- and $r$-modes are non-linearly excited by the OsC modes and whether or not the amplitude evolutions tend toward a state of finite amplitudes. We find that the non-linear three-mode couplings do excite low frequency $g$- and $r$-modes but they are not necessarily effective to achieve amplitude saturation since the three-mode couplings between the OsC modes with large growth rates and $g$- and $r$-modes with small damping rates tend to destabilize amplitude evolutions.

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arXiv2021

Overstable Convective Modes in Rotating Early Type Stars

We calculate overstable convective (OsC) modes of $2M_\odot$, $4M_\odot$, and $20M_\odot$ main sequence stars. To compute non-adiabatic OsC modes in the core, we assume $(\nabla\cdot\vec{F}_C)^\prime=0$ as a prescription for the approximation called frozen-in convection in pulsating stars where $\vec{F}_C$ is the convective energy flux and the prime $^\prime$ indicates Eulerian perturbation. We find that the general properties of the OsC modes are roughly the same as those obtained by Lee \& Saio (2020) who assumed $δ(\nabla\cdot\vec{F}_C)=0$, except that no OsC modes behave like inertial modes when they tend toward complete stabilization with increasing rotation frequency where $δ$ indicates the Lagrangian perturbation. As the rotation frequency of the stars increases, the OsC modes are stabilized to resonantly excite $g$-modes in the envelope when the core rotates slightly faster than the envelope. The frequency of the OsC modes that excite envelope $g$-modes is approximately given by $σ\sim |mΩ_c|$ in the inertial frame and hence $σ_{m=-2}\approx2σ_{m=-1}$ where $m$ is the azimuthal wavenumber of the modes and $Ω_c$ is the rotation frequency of the core. We find that the modal properties of OsC modes do not strongly depend on the mass of the stars. We discuss angular momentum transport by OsC modes in resonance with envelope $g$-modes in the main sequence stars. We suggest that angular momentum transfer takes place from the core to the envelope and that the OsC modes may help the stars rotate uniformly and keep the rotation frequency of the core low during their evolution as main sequence stars.

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arXiv2021

Rotation of the convective core in $γ$ Dor stars measured by dips in period spacings of g modes coupled with inertial modes

The relation of period spacing ($ΔP$) versus period ($P$) of dipole prograde g modes is known to be useful to measure rotation rates in the g-mode cavity of rapidly rotating $γ$ Dor and slowly pulsating B (SPB) stars. In a rapidly rotating star, an inertial mode in the convective core can resonantly couple with g modes propagative in the surrounding radiative region. The resonant coupling causes a dip in the $P$-$ΔP$ relation, distinct from the modulations due to the chemical composition gradient. Such a resonance dip in $ΔP$ of prograde dipole g modes appears around a frequency corresponding to a spin parameter $2f_{\rm rot}{\rm(cc)}/ν_{\rm co-rot} \sim 8-11$ with $f_{\rm rot}$(cc) being the rotation frequency of the convective core and $ν_{\rm co-rot}$ the pulsation frequency in the co-rotating frame. The spin parameter at the resonance depends somewhat on the extent of core overshooting, central hydrogen abundance, and other stellar parameters. We can fit the period at the observed dip with the prediction from prograde dipole g modes of a main-sequence model, allowing the convective core to rotate differentially from the surrounding g-mode cavity. We have performed such fittings for 16 selected $γ$ Dor stars having well defined dips, and found that the majority of $γ$ Dor stars we studied rotate nearly uniformly, while convective cores tend to rotate slightly faster than the g-mode cavity in less evolved stars.

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arXiv2020

Rotating Convective Core Excites Non-Radial Pulsations to Cause Rotational Modulations in Early-Type Stars

We discuss low-frequency g modes excited by resonant couplings with weakly unstable oscillatory convective modes in the rotating convective core in early-type main-sequence stars. Our non-adiabatic pulsation analyses including the effect of Coriolis force for $2\,M_\odot$ main-sequence models show that if the convective core rotates slightly faster than the surrounding radiative layers, g modes in the radiative envelope are excited by a resonance coupling. The frequency of the excited g mode in the inertial frame is close to $|mΩ_{\rm c}|$ with $m$ and $Ω_{\rm c}$ being the azimuthal order of the g mode and the rotation frequency of the convective core, respectively. These g mode frequencies are consistent with those of photometric rotational modulations and harmonics observed in many early-type main-sequence stars.

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arXiv2020

Tidal Oscillations of Rotating Hot Jupiters

We calculate small amplitude gravitational and thermal tides of uniformly rotating hot Jupiters composed of a nearly isentropic convective core and a geometrically thin radiative envelope. We treat the fluid in the convective core as a viscous fluid and solve linearized Navier Stokes equations to obtain tidal responses of the core, assuming that the Ekman number ${\rm Ek}$ is a constant parameter. In the radiative envelope, we take account of the effects of radiative dissipations on the responses. The properties of tidal responses depend on thermal timescales $τ_*$ in the envelope and Ekman number Ek in the core and on weather the forcing frequency $ω$ is in the inertial range or not, where the inertial range is defined by $|ω|\le2Ω$ for the rotation frequency $Ω$. If ${\rm Ek}\gtrsim 10^{-7}$, the viscous dissipation in the core is dominating the thermal contributions in the envelope for $τ_*\gtrsim 1$ day. If ${\rm Ek}\lesssim 10^{-7}$, however, the viscous dissipation is comparable to or smaller than the thermal contributions and the envelope plays an important role to determine the tidal torques. If the forcing is in the inertial range, frequency resonance of the tidal forcing with core inertial modes significantly affects the tidal torques, producing numerous resonance peaks of the torque. Depending on the sign of the torque in the peaks, we suggest that there exist cases in which the resonance with core inertial modes hampers the process of synchronization between the spin and orbital motion of the planets.

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arXiv2019

Thermal Tides in Rotating Hot Jupiters

We calculate tidal torque due to semi-diurnal thermal tides in rotating hot Jupiters, taking account of the effects of radiative cooling in the envelope and of the planets rotation on the tidal responses. We use a simple Jovian model composed of a nearly isentropic convective core and a thin radiative envelope. To represent the tidal responses of rotating planets, we employ series expansions in terms of spherical harmonic functions $Y_l^m$ with different $l$s for a given $m$. For low forcing frequency, there occurs frequency resonance between the forcing and the $g$- and $r$-modes in the envelope and inertial modes in the core. We find that the resonance enhances the tidal torque, and that the resonance with the $g$- and $r$-modes produces broad peaks and that with the inertial modes very sharp peaks, depending on the magnitude of the non-adiabatic effects associated with the oscillation modes. We also find that the behavior of the tidal torque as a function of the forcing frequency (or period) is different between prograde and retrograde forcing, particularly for long forcing periods because the $r$-modes, which have long periods, exist only on the retrograde side.

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arXiv2018

Overstable Convective Modes of Rotating Hot Jupiters

We calculate overstable convective modes of uniformly rotating hot Jupiters, which have a convective core and a thin radiative envelope. Convective modes in rotating planets have complex frequency $ω$ and are stabilized by rapid rotation so that their growth rates $\proptoω_{\rm I}={\rm Im}(ω)$ are much smaller than those for the non-rotating planets. The stabilized convective modes excite low frequency gravity waves in the radiative envelope by frequency resonance between them. We find that the convective modes that excite envelope gravity waves remain unstable even in the presence of non-adiabatic dissipations in the envelope. We calculate the heating rates due to non-adiabatic dissipations of the oscillation energy of the unstable convective modes and find that the magnitudes of the heating rates cannot be large enough to inflate hot Jupiters sufficiently so long as the oscillation amplitudes remain in the linear regime.

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arXiv2018

Axisymmetric magnetic modes of neutron stars having mixed poloidal and toroidal magnetic fields

We calculate axisymmetric magnetic modes of a neutron star possessing a mixed poloidal and toroidal magnetic field, where the toroidal field is assumed to be proportional to a dimensionless parameter $ζ_0$. Here, we assume an isentropic structure for the neutron star and consider no effects of rotation. Ignoring the equilibrium deformation due to the magnetic field, we employ a polytrope of the index $n=1$ as the background model for our modal analyses. For the mixed poloidal and toroidal magnetic field with $ζ_0\not=0$, axisymmetric spheroidal and toroidal modes are coupled. We compute axisymmetric spheroidal and toroidal magnetic modes as a function of the parameter $ζ_0$ from $0$ to $\sim 1$ for the surface field strengths $B_S=10^{14}$G and $10^{15}$G. We find that the frequency $ω$ of the magnetic modes decreases with increasing $ζ_0$. We also find that the frequency of the spheroidal magnetic modes is almost exactly proportional to $B_S$ for $ζ_0\lesssim 1$ but that this proportionality holds only when $ζ_0\ll 1$ for the toroidal magnetic modes. The wave patterns of the spheroidal magnetic modes and toroidal magnetic modes are not strongly affected by the coupling so long as $ζ_0\lesssim 1$. We find no unstable modes having $ω^2<0$.

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arXiv2017

Theory and evidence of global Rossby waves in upper main-sequence stars: r-mode oscillations in many Kepler stars

Asteroseismic inference from pressure modes (p modes) and buoyancy, or gravity, modes (g modes) is ubiquitous for stars across the Hertzsprung--Russell diagram. Until now, however, discussion of r modes (global Rossby waves) has been rare. Here we derive the expected frequency ranges of r modes in the observational frame by considering the visibility of these modes. We find that the frequencies of r modes of azimuthal order $m$ appear as groups at slightly lower frequency than $m$ times the rotation frequency. Comparing the visibility curves for r modes with Fourier amplitude spectra of Kepler light curves of upper main-sequence B, A and F stars, we find that r modes are present in many $γ$ Dor stars (as first discovered by Van Reeth et al. 2016), spotted stars, and so-called Heartbeat stars, which are highly eccentric binary stars. We also find a signature of r modes in a frequently bursting Be star observed by Kepler. In the amplitude spectra of moderately to rapidly rotating $γ$ Dor stars, r-mode frequency groups appear at lower frequency than prograde g-mode frequency groups, while in the amplitude spectra of spotted early A to B stars, groups of symmetric (with respect to the equator) r-mode frequencies appear just below the frequency of a structured peak that we suggest represents an approximate stellar rotation rate. In many Heartbeat stars, a group of frequencies can be fitted with symmetric $m=1$ r modes, which can be used to obtain rotation frequencies of these stars.

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arXiv2017

Axisymmetric spheroidal modes of neutron stars magnetized with poloidal magnetic fields

We calculate axisymmetric spheroidal modes of neutron stars magnetized with an axisymmetric poloidal magnetic field. We use polytropes of the indices $n\sim1$ as background equilibrium models of neutron stars where we ignore the deformation due to the magnetic fields. For a poloidal magnetic field, axisymmetric normal modes of non-rotating stars are decoupled into spheroidal modes and toroidal modes, and we can treat spheroidal modes separately from toroidal modes. For the surface field strength $B_S$ ranging from $10^{14}$G to $10^{16}$G, we calculate axisymmetric spheroidal magnetic modes whose oscillation frequency is proportional to $B_S$. The typical oscillation frequency of the magnetic modes is $\sim 0.01\times\sqrt{GM/R^3}$ for $B_S\sim 10^{15}$G, where $M$ and $R$ are respectively the mass and radius of the star and $G$ is the gravitational constant. For $M=1.4M_\odot$ and $R=10^6$cm, this frequency is $\sim 20$Hz, which may explain low frequency QPOs found for SGR 1806-204 and SGR 1900+14. We also find modes of frequency $> \sqrt{GM/R^3}$ corresponding to the radial fundamental and first harmonic modes. No unstable magnetic modes are found for axisymmetric spheroidal oscillations of magnetized stars.

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arXiv2017

Tidally driven mean flows in slowly and uniformly rotating massive main sequence stars

We calculate tidally driven mean flows in a slowly and uniformly rotating massive main sequence star in a binary system. We treat the tidal potential due to the companion as a small perturbation to the primary star. We compute tidal responses of the primary as forced linear oscillations, as a function of the tidal forcing frequency $ω_{\rm tide}=2(Ω_{\rm orb}-Ω)$, where $Ω_{\rm orb}$ is the mean orbital angular velocity and $Ω$ is the angular velocity of rotation of the primary star. The amplitude of the tidal responses is proportional to the parameter $f_0\propto (M_2/M)(a_{\rm orb}/R)^{-3}$, where $M$ and $M_2$ are the masses of the primary and companion stars, $R$ is the radius of the primary and $a_{\rm orb}$ is the mean orbital separation between the stars. For a given $f_0$, the amplitudes depend on $ω_{\rm tide}$ and become large when $ω_{\rm tide}$ is in resonance with natural frequencies of the star. Using the tidal responses, we calculate axisymmetric mean flows, assuming that the mean flows are non-oscillatory flows driven via non-linear effects of linear tidal responses. We find that the $ϕ$-component of the mean flow velocity dominates. We also find that the amplitudes of the mean flows are large only in the surface layers where non-adiabatic effects are significant and that the amplitudes are confined to the equatorial regions of the star. Depending on $M_2/M$ and $a_{\rm orb}/R$, the amplitudes of mean flows at the surface become significant.

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arXiv2016

Pulsation-driven mean zonal and meridional flows in rotating massive stars

Zonal and meridional axisymmetric flows can deeply impact the rotational and chemical evolution of stars. Therefore, momentum exchanges between waves propagating in stars, differential rotation, and meridional circulation must be carefully evaluated. In this work, we study axisymmetric mean flows in rapidly and initially uniformly rotating massive stars driven by small amplitude non-axisymmetric $κ$-driven oscillations. We treat them as perturbations of second-order of the oscillation amplitudes and derive their governing equations as a set of coupled linear ordinary differential equations. This allows us to compute 2-D zonal and meridional mean flows driven by low frequency $g$- and $r$-modes in slowly pulsating B (SPB) stars and $p$-modes in $β$ Cephei stars. Oscillation-driven mean flows usually have large amplitudes only in the surface layers. In addition, the kinetic energy of the induced 2-D zonal rotational motions is much larger than that of the meridional motions. In some cases, meridional flows have a complex radial and latitudinal structure. For SPB stars, we find that there are low frequency retrograde $g$-modes and $r$-modes that drive mean flows with positive velocity components in the radial and azimuthal directions at the stellar surface. This suggests that low frequency retrograde modes can transport angular momentum to the surface layers, which may help to form a circumstellar gaseous disc. Moreover, pulsation-driven and rotation-driven meridional flows can have similar amplitudes. These results show the importance of taking wave-- mean flow interactions into account when studying the evolution of massive stars.

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arXiv2015

Non-axisymmetric magnetic modes of neutron stars with purely poloidal magnetic fields

We calculate non-axisymmetric oscillations of neutron stars magnetized by purely poloidal magnetic fields. We use polytropes of index $n=1$ and 1.5 as a background model, where we ignore the equilibrium deformation due to the magnetic field. Since separation of variables is not possible for the oscillation of magnetized stars, we employ finite series expansions for the perturbations using spherical harmonic functions. Solving the oscillation equations as the boundary and eigenvalue problem, we find two kinds of discrete magnetic modes, that is, stable (oscillatory) magnetic modes and unstable (monotonically growing) magnetic modes. For isentropic models, the frequency or the growth rate of the magnetic modes is exactly proportional to $B_{\rm S}$, the strength of the field at the surface. The oscillation frequency and the growth rate are affected by the buoyant force in the interior, and the stable stratification tends to stabilize the unstable magnetic modes.

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arXiv2015

Non-radial oscillations of the magnetized rotating stars with purely toroidal magnetic fields

We calculate non-axisymmetric oscillations of uniformly rotating polytropes magnetized with a purely toroidal magnetic field, taking account of the effects of the deformation due to the magnetic field. As for rotation, we consider only the effects of Coriolis force on the oscillation modes, ignoring those of the centrifugal force, that is, of the rotational deformation of the star. Since separation of variables is not possible for the oscillation of rotating magnetized stars, we employ finite series expansions for the perturbations using spherical harmonic functions. We calculate magnetically modified normal modes such as $g$-, $f$-, $p$-, $r$-, and inertial modes. In the lowest order, the frequency shifts produced by the magnetic field scale with the square of the characteristic Alfvén frequency. As a measure of the effects of the magnetic field, we calculate the proportionality constant for the frequency shifts for various oscillation modes. We find that the effects of the deformation are significant for high frequency modes such as $f$- and $p$-modes but unimportant for low frequency modes such as $g$-, $r$-, and inertial modes.

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arXiv2014

Axisymmetric toroidal modes of general relativistic magnetized neutron star models

We calculate axisymmetric toroidal modes of magnetized neutron stars with a solid crust in the general relativistic Cowling approximation. We assume that the interior of the star is threaded by a poloidal magnetic field, which is continuous at the surface with an outside dipole field. We examine the cases of the field strength $B_{\rm{S}}\sim10^{16}$ G at the surface. Since separation of variables is not possible for the oscillations of magnetized stars, we employ finite series expansions for the perturbations using spherical harmonic functions. We find discrete normal toroidal modes of odd parity, but no toroidal modes of even parity are found. The frequencies of the toroidal modes form distinct mode sequences and the frequency in a given mode sequence gradually decreases as the number of radial nodes of the eigenfunction increases. From the frequency spectra computed for neutron stars of different masses, we find that the frequency is almost exactly proportional to $B_{\rm{S}}$ and is well represented by a linear function of $R/M$ for a given $B_{\rm{S}}$, where $M$ and $R$ are the mass and radius of the star. The toroidal mode frequencies for $B_{\rm{S}}\sim 10^{15}$ G are in the frequency range of the quasi-periodic oscillations (QPOs) detected in the soft-gamma-ray repeaters, but we find that the toroidal normal modes cannot explain all the detected QPO frequencies.

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arXiv2014

Angular momentum transport by stochastically excited oscillations in rapidly rotating massive stars

We estimate the amount of angular momentum transferred by the low-frequency oscillations detected in the rapidly rotating hot Be star HD 51452. Here, we assume that the oscillations detected are stochastically excited by convective motions in the convective core of the star, that is, we treat the oscillations as forced oscillations excited by the periodic convective motions of the core fluids having the frequencies observationally determined. With the observational amplitudes of the photometric variations, we determine the oscillation amplitudes, which makes it possible to estimate the net amount of angular momentum transferred by the oscillations using the wave-meanflow interaction theory. Since we do not have any information concerning the azimuthal wavenumber $m$ and spherical harmonic degree $l$ for each of the oscillations, we assume that all the frequencies detected are prograde or retrograde in the observer's frame and they are all associated with a single value of $m$ both for even modes ($l=|m|$) and for odd modes ($l=|m|+1$). We estimate the amount of angular momentum transferred by the oscillations for $|m|=1$ and 2, which are typical $|m|$ values for Be stars, and find that the amount is large enough for a decretion disc to form around the star. Therefore, transport of angular momentum by waves stochastically excited in the core of Be stars might be responsible for the Be phenomenon.

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arXiv2014

Excitation of a nonradial mode in a millisecond X-ray pulsar XTE J1751-305

We discuss candidates for non-radial modes excited in a mass accreting and rapidly rotating neutron star to explain the coherent frequency identified in the light curves of a millisecond X-ray pulsar XTE J1751-305. The spin frequency of the pulsar is $ν_{\rm spin}\cong435$Hz and the identified coherent frequency is $ν_{\rm osc}=0.5727595\timesν_{\rm spin}$. Assuming the frequency identified is that observed in the corotating frame of the neutron star, we find that the surface $r$-modes of $l^\prime=m=1$ and 2 excited by $ε$-mechanism due to helium burning in the thin shell can give the frequency ratio $κ=ν_{\rm osc}/ν_{\rm spin}\simeq0.57$ at $ν_{\rm spin}=435$Hz. As another candidate for the observed ratio $κ$, we also suggest a toroidal crustal mode that has penetrating amplitudes in the fluid core and is destabilized by gravitational wave emission. Since the surface fluid layer is separated from the fluid core by a solid crust, the amplitudes of an $r$-mode in the core, which is destabilized by emitting gravitational waves, can be by a large factor different from those in the fluid ocean. We find that the amplification factor defined as $f_{\rm amp}=α_{\rm surface}/α_{\rm core}$ is as large as $f_{\rm amp}\sim 10^2$ for the $l^\prime=m=2$ $r$-mode at $ν_{\rm spin}=435$Hz for a $M=1.4M_\odot$ neutron star model. Because of this significant amplification of the $r$-mode amplitudes in the surface fluid layer, we suggest that, when proper corrections to the $r$-mode frequency such as due to the general relativistic effects are taken into consideration, the core $r$-mode of $l^\prime=m=2$ can be a candidate for the detected frequency, without leading to serious contradictions to, for example, the spin evolution of the underlying neutron star.

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