Issue 
A&A
Volume 647, March 2021



Article Number  L1  
Number of page(s)  3  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/202140400  
Published online  08 March 2021 
Letter to the Editor
Starspot modelling of the TESS light curve of CVSO 30
Department of Statistics, University of the Western Cape, Private Bag X17, Bellville, 7535 Cape, South Africa
email: ckoen@uwc.ac.za
Received:
22
January
2021
Accepted:
22
February
2021
Aims. I aim to investigate whether the photometric variability in the candidate host star CVSO 30 can be explained by starspots.
Methods. The Transiting Exoplanet Survey Satellite (TESS) light curve of CVSO 30 is separated into two independent nonsinusoidal periodic components. A starspot modelling technique is applied to each of these components.
Results. Combined, the two model light curves reproduce the TESS observations to a high accuracy, obviating the need to invoke planetary transits to describe part of the variability.
Key words: stars: variables: T Tauri, Herbig Ae/Be
© 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 lowamplitude 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 latetype 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
where: ψ is the rotational phase; I(ψ) and I_{0} are, respectively, the observed luminosity and the luminosity that would have been observed from an unspotted stellar surface; F_{*} and F_{s} 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
Koen (2021) showed that by discretising the integral, Eq. (2) can be approximated by the linear regression equation
In Eq. (3), y is a column vector with elements
Each entry in the vector z corresponds to a particular small surface element:
that is, z_{j} = 0 if pixel j is unspotted. Entries in the L × p matrix G depend only on the limb darkening coefficients and on the angle γ_{j} 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 normn of z, a simple constrained form of Eq. (3) is
where the constant λ determines the relative importance of the model fit and the parsimony of the nonzero 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 z_{j} (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, R^{2} measures the fraction of the observed variability of m explained by the model. In addition to R^{2}, which is not very sensitive to locally poor fits, two other useful statistics are
Values of d and σ_{ϵ} will be given as percentages of the peaktopeak 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 T_{eff} ≈ 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 ≤ z_{j} ≤ 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 function^{1}, F_{s}/F_{*} ≥ 0.49 or z_{j} = 1 − F_{s}/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 F_{s} ≠ F_{*} required to achieve the fit. The largest values that still gave excellent global fits (R^{2} = 1) were chosen. The surface of each star was subdivided into p = 20 000 elements of equal area (2.06 deg^{2}).
3. Results
TESS (Ricker et al. 2015) observed CVSO 30 for 21.8 d at a twominute cadence. The data are available from the Mikulski Archive for Space Telescopes (MAST) portal^{2}. 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 norm1 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 norm2 models. Spot patterns can be seen in Figs. 2 and 3.
Fig. 1. Norm1 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. 
Details of the spot models.
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. 
Fig. 3. Same as for Fig. 2, but for norm2 models. The different intensity scales in the two panels should be noted. 
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.
Fig. 4. Twominute 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. 
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). Multifilter 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.
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All Tables
All Figures
Fig. 1. Norm1 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. 

In the text 
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. 

In the text 
Fig. 3. Same as for Fig. 2, but for norm2 models. The different intensity scales in the two panels should be noted. 

In the text 
Fig. 4. Twominute 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. 

In the text 
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