Modeling photometric variations due to a global inhomogeneity on an obliquely rotating star application to lightcurves of white dwarfs

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Modeling photometric variations due to a global

inhomogeneity on an obliquely rotating star:

application to lightcurves of white dwarfs

Yasushi SUTO1,2,3, Shin SASAKI4, Masataka AIZAWA5, Kotaro FUJISAWA1,

and Kazumi KASHIYAMA6,7

1Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan

2Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo

113-0033, Japan

3Laboratory of Physics, Kochi University of Technology, Tosa Yamada, Kochi 782-8502, Japan

4Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

5Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shengrong Road 520, 201210

Shanghai, P. R. China

6Astronomical Institute, Tohoku University, Sendai 980-8578, Japan

7Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo,

Kashiwa 277-8583, Japan

∗E-mail: suto@phys.s.u-tokyo.ac.jp

Received 2022 October 3; Accepted 2022 November 7

Abstract

We develop a general framework to compute photometric variations induced by the oblique

rotation of a star with an axisymmetric inhomogeneous surface. We apply the framework to

compute lightcurves of white dwarfs adopting two simple models of their surface inhomogene-

ity. Depending on the surface model and the location of the observer, the resulting lightcurve

exhibits a departure from a purely sinusoidal curve that are observed for a fraction of white

dwarfs. As a speciﬁc example, we ﬁt our model to the observed phase-folded lightcurve of

a fast-spinning white dwarf ZTF J190132.9+145808.7 (with the rotation period of 419s). We

ﬁnd that the size and obliquity angle of the spot responsible for the photometric variation are

arXiv:2210.05421v2 [astro-ph.SR] 7 Nov 2022

∆θs≈60◦and θ?≈60◦or 90◦, respectively, implying an interesting constraint on the surface

distribution of the magnetic ﬁeld on white dwarfs.

Key words: stars:rotation – starspots – white dwarfs

1 Introduction

A number of white dwarfs (WDs) have exhibited periodic photometric variations (e.g., Achilleos

et al. 1992; Barstow et al. 1995; Wade et al. 2003; Brinkworth et al. 2005, 2013; Reding et al.

2020; Caiazzo et al. 2021; Kilic et al. 2021; Williams et al. 2022). Those variations are commonly

interpreted to originate from inhomogeneities on the stellar surface that obliquely rotate around

the spin axis. Those inhomogeneities may be produced by magnetic spots in a convective

atmosphere (e.g., Valyavin et al. 2008, 2011) or the so-called magnetic dichroism (Ferrario

et al. 1997); the continuum opacity changes with rotational phase according to the amplitude

of the magnetic ﬁeld strength across the stellar disk. As a result, the photometric ﬂux varies

as well. Hence, photometric lightcurves of WDs carry rich information on the distribution

of the surface magnetic ﬁelds, which may be further disentangled by combining the spectro-

polarimetric data if available (e.g., Liebert et al. 1977; Donati et al. 1994; Euchner et al. 2002;

Valyavin et al. 2014).

For instance, Brinkworth et al. (2013) have performed time-series photometry of 30 iso-

lated magnetic WDs, and 5 (24 %) are variable with reliably measured periods of (1 – 7) hours.

Quite interestingly, all of their lightcurves are well ﬁtted by a monochromatic sinusoidal curve.

Indeed, apart from 9 WDs whose lightcurves are signiﬁcantly contaminated by their varying

comparison stars, they ﬁnd that 14 out of the remaining 21 show evidence for variability whose

lightcurves are consistent with a monochromatic sinusoidal curve. In marked contrast to other

stars for which non-sinusoidal rotation signatures are commonly observed (e.g., Roettenbacher

et al. 2013), the sinusoidal lightcurve seems to be fairly generic for these types of WDs, which

have a relatively stable surface structure over time.

Thanks to the Kepler satellite, Transiting Exoplanet Survey Satellite (TESS), and

Zwicky Transient Facility (ZTF) (Maoz et al. 2015; Reding et al. 2020; Hermes et al. 2021)

among others, the number of photometric lightcurves for WDs with a high accuracy and ca-

dence is rapidly increasing. Indeed, fast-spinning WDs with a rotation period even less than 10

minutes have been recently discovered (Reding et al. 2020; Caiazzo et al. 2021; Kilic et al. 2021).

For such cases, however, it may be due to their pulsation, instead of the rotation, since their

frequency ranges are overlapped (e.g., Winget & Kepler 2008; Hollands et al. 2020). This points

to an importance of quantitative modeling of photometric lightcurves due to inhomogeneities

on rotating stars. For instance, the phase-folded lightcurve of one of the fast-spinning WDs

(ZTF J190132.9+145808.7) seems to exhibit a non-sinusoidal feature (Caiazzo et al. 2021, and

see Sec. 3.5 below), which could be relatively common for these young massive WDs. Thus,

such modeling of photometric variations due to an inhomogeneity on the stellar surface should

become even more crucial in the coming era of the Rubin Observatory LSST Camera (Ivezi´c

et al. 2019).

Photometric rotational variations due to multiple circular starspots have been studied

previously (e.g., Budding 1977; Dorren 1987; Eker 1994; Landolﬁ et al. 1997; Kipping 2012).

More recently, Suto et al. (2022) proposed a fully analytic model for the photometric variations

due to inﬁnitesimally small multiple starspots on a diﬀerentially rotating star, and performed a

series of mock Lomb-Scargle analysis relevant to the Kepler data (e.g., Lu et al. 2022). In this

paper, instead, we focus on an inhomogeneous surface intensity pattern that is axisymmetric

around an axis misaligned to the stellar spin axis. The inhomogeneity produces periodic pho-

tometric variations due to the stellar rotation. The purpose of the present paper is to develop

a general formulation to describe the rotational variation, compute the resulting lightcurves for

a couple of physically-motivated speciﬁc models for WDs, and apply the methodology to derive

constraints on several parameters for ZTF J190132.9+145808.7 as an example.

The rest of the paper is organized as follows. Section 2 presents our formulation of

the lightcurve modeling for a solid-body rotating star with an inhomogeneous surface intensity

distribution. We ﬁrst present a general formulation to compute the photometric modulation

due to the oblique stellar rotation by adopting a quadratic limb darkening law. While our

formulation would be essentially the same as those in the previous literature, the resulting

expressions are characterized by a set of parameters with clear physical interpretations, and

thus more useful in ﬁtting to the observed lightcurves. In order to show the advantage of our

method, we apply the formulation to two simple inhomogeneity models in section 3: a constant-

intensity single circular spot (cap-model), and a globally varying-intensity surface (p-model).

We ﬁnd that the p-model leads to a sinusoidal lightcurve strictly, while the cap-model exhibits

a departure from a sinusoidal lightcurve depending on the geometrical conﬁgurations between

the starspot and the observer. In both models, the photometric variations are fairly insensitive

to the limb darkening eﬀect. In subsection 3.5, we show that the cap-model well explains the

observed lightcurve of ZTF J190132.9+145808.7. Finally, section 4 is devoted to discussion and

conclusion of the present paper. Analytic derivations of several integrals appearing in the main

text are given in Appendix.

2 Basic formulation of the lightcurve modeling for an obliquely rotating star

As shown in Figure 1, we consider a star rotating along the z-axis with a spin angular frequency

of ω?, and the observer is located at (sinθo,0,cosθo). If we denote the obliquity angle between

the stellar spin axis and the symmetry axis of the surface inhomogeneity distribution by θ?, the

angle between the observer’s line-of-sight and the symmetry axis, γ(t) is given by

cosγ(t) = sinθ?sinθocosω?t+ cosθ?cosθo,(1)

and thus γ(t) varies between |θ?−θo|and θ?+θo(< π). In practice, the symmetry axis may

correspond to the magnetic dipole axis of WDs, or to the central axis of a single spherical spot

on the stellar surface. While we assume that θ?is constant throughout the present analysis,

its possible time-dependence is easily incorporated by substituting the speciﬁc function θ?(t)

in equation (1) because the photometric variation in our formulation is completely speciﬁed by

γ(t) alone.

Following Suto et al. (2022), the normalized photometric lightcurve of the stellar surface

L(t) = ZZ K(θ,ϕ)I(θ,ϕ)ILD(θ, ϕ) sinθ dθ dϕ

ZZ K(θ,ϕ)Isinθ dθ dϕ

πI ZZ K(θ, ϕ)I(θ, ϕ)ILD(θ,ϕ)sinθ dθ dϕ, (2)

where Kis the weighting kernel of the surface visible to the observer, I(θ, ϕ) and ILD(θ, ϕ)

indicate the surface intensity distribution and the limb darkening (the edge of the stellar disk is

observed to be dimmer than its central part), respectively, and the integration is performed over

the entire stellar surface (see also, Fujii et al. 2010, 2011; Farr et al. 2018; Haggard & Cowan

2018; Nakagawa et al. 2020). We introduce a constant surface intensity, I, just for normalization,

but it is not directly determined from observed data and can be chosen arbitrarily.

For an isotropically emitting stellar surface, the weighting kernel Kis equivalent to the

visibility computed from the direction cosine between the unit normal vector of the stellar

surface (e?) and the unit vector toward the observer (eo). Without loss of generality, one can

deﬁne a spherical coordinate system in which the symmetry axis is instantaneously set to be

z-axis. In this case, one obtains

e?·eo≡µ?= sinγ(t)sinθ[cosϕ+ Γ(t)],(3)

Fig. 1. Schematic illustration of the star and the observer. The observer is located at (sin θo,0,cos θo). In this frame, the stellar spin axis is chosen as the

Z-axis, and the unit vector of symmetry axis is deﬁned to be (sin θ?,0,cos θ?)at t= 0.

where

Γ(t)≡cotγ(t)cotθ. (4)

The location (θ,ϕ) on the surface is visible to the observer if µ?>0. Therefore, the weighting

kernel is simply written as

K(θ,ϕ) = max(µ?,0) = sinγ(t)sinθmax[cosϕ+ Γ(t),0].(5)

If we adopt a quadratic limb darkening law, ILD in equation (2) is speciﬁed by the two param-

eters u1and u2as

ILD(θ, ϕ)=1−u1(1 −µ?)−u2(1 −µ?)2= (1 −u1−u2) + (u1+ 2u2)µ?−u2µ2

?.(6)

For instance, u1= 0.47 and u2= 0.23 for the Sun at 550nm, and u1= 0.05 and u2= 0.51 for a

typical white dwarf with Teﬀ = 104K and log g=8.0 in the LSST g-band (Cox 2000; Gianninas

et al. 2013).

We focus on a case where the surface intensity distribution is axisymmetric, i.e., I(θ,ϕ) =

I(θ). Then, equation (2) reduces to

L(t) = L0(t) + L1(t) + L2(t),(7)

where

L0(t) = (1 −u1−u2)sinγ

πI ZZ max[cosϕ+ Γ(t),0] sin2θ I(θ)dθdϕ, (8)

L1(t)=(u1+ 2u2)sin2γ

πI ZZ max[cosϕ+ Γ(t),0] sin3θ[cosϕ+ Γ(t)]I(θ)dθdϕ, (9)

and

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Modelingphotometricvariationsduetoaglobalinhomogeneityonanobliquelyrotatingstar:applicationtolightcurvesofwhitedwarfsYasushiSUTO1,2,3,ShinSASAKI4,MasatakaAIZAWA5,KotaroFUJISAWA1,andKazumiKASHIYAMA6,71DepartmentofPhysics,TheUniversityofTokyo,Tokyo113-0033,Japan2ResearchCenterfortheEarlyUniverse,Schoo...

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Modeling photometric variations due to a global inhomogeneity on an obliquely rotating star application to lightcurves of white dwarfs

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