Free Access
Issue
A&A
Volume 647, March 2021
Article Number L1
Number of page(s) 3
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202140400
Published online 08 March 2021

© ESO 2021

1. Introduction

The T Tauri star CVSO 30 was discovered by Briceño et al. (2005). Van Eyken et al. (2012) analysed extensive Palomar Transit Factory observations of the star and found periodic low-amplitude depressions in the light curve with durations of ∼2 h. These were ascribed to planetary transits. A second planetary candidate (CVSO 30c), found by direct imaging, was announced by Schmidt et al. (2016) (but see Lee & Chiang 2018, for a different interpretation). The possible association of planets with CVSO 30 is of particular interest due to the young age of the star (∼2.6 Myr; Briceño et al. 2005). Other authors who have obtained further observations of periodic flux dips, or who proposed detailed models for these, include Raetz et al. (2016), Barnes et al. (2013), Kamiaka et al. (2015), Yu et al. (2015), Koen (2015), Howarth (2016), Onitsuka et al. (2017), and Tanimoto et al. (2020).

Koen (2015) pointed out that the periodic flux dips discovered by Van Eyken et al. (2012) may simply be part of a complex variation pattern of the star. This theme was elaborated on by Koen (2020) and Bouma et al. (2020), who independently came to the conclusion that the star was a binary T Tauri with no overt planetary companions. The light variations of both stars are thought be complex, showing dips. This Letter confirms that finding and presents models for both sets of variations.

A variety of explanations for complex light curves in young late-type stars have been discussed in the literature, for example in Stauffer et al. (2017, 2018), Zhan et al. (2019), and Günther et al. (2020). Here, it is shown that the variations could be due to dark features on the surfaces of the stars. The modelling methodology is very similar to that in Koen (2021).

2. The starspot model

The fundamental equations describing the flux variations due to the rotation of a spotted star are

(1)

where: ψ is the rotational phase; I(ψ) and I0 are, respectively, the observed luminosity and the luminosity that would have been observed from an unspotted stellar surface; F* and Fs are, respectively, the fluxes from unspotted and spotted surface areas; the function h(μ) describes limb darkening; and μ = cos γ, where γ is the angle between the surface normal and the line of sight towards the observer (see e.g., Dorren 1987, and references therein). Equation (1) is a slight generalisation of Eq. (1) in Koen (2021) to quadratic, rather than linear, limb darkening; regarding the usual coefficients a and b (e.g., Claret 2017),

It follows from this Letter’s Eq. (1) that

(2)

Koen (2021) showed that by discretising the integral, Eq. (2) can be approximated by the linear regression equation

(3)

In Eq. (3), y is a column vector with elements

Each entry in the vector z corresponds to a particular small surface element:

(4)

that is, zj = 0 if pixel j is unspotted. Entries in the L × p matrix G depend only on the limb darkening coefficients and on the angle γj between the local surface normal and the line of sight. The vector e models noise.

Constraints need to be placed on z in order to make the regression problem tractable. Using the notation

for the norm-n of z, a simple constrained form of Eq. (3) is

(5)

where the constant λ determines the relative importance of the model fit and the parsimony of the non-zero components of z. The cases n = 1 and n = 2 are known, respectively, as ‘lasso’ and ‘ridge’ regression in the literature (e.g., Hastie et al. 2009). Both lasso and ridge regression favour small values of zj (i.e. small deviations from the undisturbed photospheric flux). In the case of lasso regression, a premium is placed on zero values, that is, the regression can be seen as providing an optimally parsimonious model with a minimum number of pixels that have depressed fluxes. The quadratic programming problem (5) can easily be solved using software such as CVX (e.g., Grant & Boyd 2020).

A standard regression goodness of fit statistic is the ‘coefficient of determination’,

where

m(ψ) and being, respectively, the observed and the predicted magnitudes at phase ψ. Here, R2 measures the fraction of the observed variability of m explained by the model. In addition to R2, which is not very sensitive to locally poor fits, two other useful statistics are

(6)

Values of d and σϵ will be given as percentages of the peak-to-peak observed magnitudes in what follows.

Because of the particular formulation of the problem, only two physical quantities (the inclination angle and limb darkening) need to be specified. Results are not very sensitive to the inclination angle i; a few different values were tried, and those giving the smallest prediction errors were adopted. If it is assumed that the two stars making up the binary system are fairly similar, with Teff ≈ 3400 K (Briceño et al. 2019; Koen 2020) and log g = 4.5, then typical limb darkening coefficients for the Transiting Exoplanet Survey Satellite (TESS) filter are a = 0.18, b = 0.44 (Claret 2017).

Clearly, for dark starspots, 0 ≤ zj ≤ 1 for all j. The upper limit can be sharpened by noting that for a stellar temperature ∼3400 K, starspot temperatures will generally be about 400 K lower (see Fig. 7 in Berdyugina 2005). Assuming blackbody radiation and using the TESS filter transmission function1, Fs/F* ≥ 0.49 or zj = 1 − Fs/F* ≤ 0.51.

One mathematical constant, λ, must be specified. This parameter controls the tradeoff between the goodness of fit of the model and the number of surface elements with Fs ≠ F* required to achieve the fit. The largest values that still gave excellent global fits (R2 = 1) were chosen. The surface of each star was subdivided into p = 20 000 elements of equal area (2.06 deg2).

3. Results

TESS (Ricker et al. 2015) observed CVSO 30 for 21.8 d at a two-minute cadence. The data are available from the Mikulski Archive for Space Telescopes (MAST) portal2. Koen (2020) and Bouma et al. (2020) argue that the light curve is the sum of contributions from two different stars. Because they are periodic, Fourier methods make short work of separating the two individual light curves. The larger amplitude variation has a base frequency of 2.0038 d−1 (period 0.49905 d), with at least two harmonics also evident in a periodogram of the observations. The second light curve is much more complex, showing power up to the seventh harmonic of the fundamental frequency 2.2292 d−1 (period 0.44859 d).

The two light curves are plotted in Fig. 1 (solid lines). The figure also shows norm-1 model fits (dots), which are indistinguishable from the observations on the scales of the diagram. Details of the two models are listed in Table 1, which also gives parameters for norm-2 models. Spot patterns can be seen in Figs. 2 and 3.

thumbnail Fig. 1.

Norm-1 model fits (dots) to the phased TESS light curves of the two components of CVSO 30 (solid lines). Top panel: 0.4991 d periodicity. Bottom panel: 0.4486 d periodicity.

Open with DEXTER

Table 1.

Details of the spot models.

thumbnail Fig. 2.

Spot configurations giving rise to the model light curves in Fig. 1 (see Table 1 for further details). The colour coding indicates the ratio of local flux to the unspotted photospheric flux, i.e. it equals zero (one) for a completely dark (unspotted) area.

Open with DEXTER

thumbnail Fig. 3.

Same as for Fig. 2, but for norm-2 models. The different intensity scales in the two panels should be noted.

Open with DEXTER

A comparison of the combined model light curves and the TESS observations is given in Fig. 4. It is clear that the overall description of the data is very good. The residuals are dominated by two short stretches of systematic deviations visible towards the ends of the second and the last panels of Fig. 4. If they are excluded (spans of ∼0.15 and ∼0.47 d, respectively), the largest peak in the residual amplitude spectrum has a height of 2.8 millimagnitudes (at a period ∼7.4 d). This can most likely be ascribed to systematics in the TESS observations and/or typical T Tauri variability.

thumbnail Fig. 4.

Two-minute TESS measurements of CVSO 30 (shown with dots) and the sum of the two model light curves plotted in Fig. 1 (shown with the solid line). The width and height of each panel are 5.7 d and 0.24 mag, respectively. The total duration of the TESS run was 21.8 d.

Open with DEXTER

4. Conclusions

It should be noted that starspot models based on photometry through a single filter are not unique, as is made abundantly clear by comparing Figs. 2 and 3 (see also e.g., Vogt 1981; Basri & Shah 2020). Multi-filter light curves and time series spectroscopy will go some way towards ameliorating this problem. Nonetheless, it has been demonstrated that starspot models have the potential to accurately reproduce the variability in CVSO 30 without invoking additional sources of variability, such as planetary transits.


Acknowledgments

The author appreciates greatly that the developers of the CVX package makes it freely available for academic use. The easy access to TESS photometry via the MAST portal is gratefully acknowledged.

References

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All Tables

Table 1.

Details of the spot models.

All Figures

thumbnail Fig. 1.

Norm-1 model fits (dots) to the phased TESS light curves of the two components of CVSO 30 (solid lines). Top panel: 0.4991 d periodicity. Bottom panel: 0.4486 d periodicity.

Open with DEXTER
In the text
thumbnail Fig. 2.

Spot configurations giving rise to the model light curves in Fig. 1 (see Table 1 for further details). The colour coding indicates the ratio of local flux to the unspotted photospheric flux, i.e. it equals zero (one) for a completely dark (unspotted) area.

Open with DEXTER
In the text
thumbnail Fig. 3.

Same as for Fig. 2, but for norm-2 models. The different intensity scales in the two panels should be noted.

Open with DEXTER
In the text
thumbnail Fig. 4.

Two-minute TESS measurements of CVSO 30 (shown with dots) and the sum of the two model light curves plotted in Fig. 1 (shown with the solid line). The width and height of each panel are 5.7 d and 0.24 mag, respectively. The total duration of the TESS run was 21.8 d.

Open with DEXTER
In the text

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