# Modelling simultaneous CoRoT and Spitzer light curves: the impact of bending waves in the inner disc

E Nagel, J Bouvier

► To cite this version:

E Nagel, J Bouvier. Modelling simultaneous CoRoT and Spitzer light curves: the impact of bending waves in the inner disc. Monthly Notices of the Royal Astronomical Society, 2023, 524 (2), pp.1997-2003. 10.1093/mnras/stad1946. hal-04961067

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# Modelling simultaneous *CoRoT* and *Spitzer* light curves: the impact of bending waves in the inner disc

E. Nagel<fnref n="1" />* and J. Bouvier<fnref n="2" />*

Accepted 2023 June 22. Received 2023 June 20; in original form 2023 February 10

## ABSTRACT

We present a physical interpretation of the amplitude of light curves of young stellar objects (YSOs) simultaneously observed in the infrared with the *Spitzer Space Telescope* and in the optical with the *CoRoT* satellite. The modelled objects come from a sample of YSOs in the star forming region NGC 2264, and we focus on a subsample showing periodic dips in their *CoRoT* light curves. The model includes emission and stellar occultation caused by a warp at the inner edge of the disc. In order to fit the observed fluxes, we add a contribution coming from the inner disc. We find that the disc flux signal has the same periodicity as the stellar rotation, with a phase offset between the stellar and the disc signal of half the period, that is, the peak of the former coincides with the minimum of the latter. This behaviour is consistent with MHD simulations where a bending wave appears opposite to the warp formed at the base of the magnetospheric stream, which is rotating with the stellar rotation period. This evidence highlights the importance of infrared observations to complement the information extracted from optical light curves to be able to characterize structures at the inner disc.

**Key words:** accretion, accretion discs – stars: pre-main-sequence.

## 1 INTRODUCTION

All objects in the Universe show variability on some scale. Light curve (LC) analysis using photometric observations in the optical with the *CoRoT* telescope (Cody et al. 2014; Stauffer et al. 2015, 2016) or with the *Kepler telescope* (Roggero et al. 2021) or observations in the infrared (IR) with the *Spitzer telescope* (Morales-Calderón et al. 2011; Cody et al. 2014) conclude that, almost any young stellar object (YSO) shows variable flux. The science program YSO Variability (YSOVAR) of the *Spitzer Space Telescope* (Morales-Calderón et al. 2011; Stauffer et al. 2015) produced the LCs of a large sample of objects in the Orion Nebula Cluster, where different physical mechanisms are required to explain the observed variability, including slow drifts in brightness, rapid chaotic flux variations, flares, periodic variations sometimes mixed with longer term changes, as well as changes in colours. The main mechanisms at the origin of the variability include changes in the accretion rate, occultation by a warped inner disc, variations in the magnetospheric accretion structure, evolution of the photospheric spots, and variable extinction (Morales-Calderón et al. 2011). The presence of several variability types led Cody et al. (2014) to define a classification system, containing LCs described as purely periodic and quasi-periodic stars, bursters, stochastic stars, eclipsing binaries, and dippers. The observations described in Alencar et al. (2010), McGinnis et al. (2015), Rice et al. (2015), and Roggero et al. (2021) among others highlight the periodic AA Tau-type or dipper stars, and this is the category we focus on this work.

The subset of dipper stars showing a periodic signal of a few days in their LCs can be interpreted from structures located at the innermost region of the disc. For this reason, this region is studied using MHD simulations of the disc interaction, with an inclined dipolar stellar magnetic field (B*) (Romanova et al. 2013, 2021). For instance, Kulkarni & Romanova (2008) concluded that at the inner edge of the disc, the Rayleigh–Taylor instability is responsible for the formation of tongues reaching the stellar surface. This mechanism forms a variable pattern of spots, where the accretion geometry shapes the hot spots distribution, which is directly related to flux variability (Romanova et al. 2004). The characterization of the magnetospheric flows in a stable (Romanova et al. 2003) or an unstable regime (Romanova, Kulkarni & Lovelace 2008) is relevant to quantify the effect caused in the LC features. Another structure arising from the star-disc interaction is a wind: X-winds (Mohanty & Shu 2008), accretion-powered stellar winds (Matt & Pudritz 2005), and magnetospheric ejections (Zanni & Ferreira 2013). These outflows shape the region above the disc and also change the characteristics of the accreting material, due to loss of angular momentum. The occurrence of all these structures formed in a highly dynamic environment contributes to the shaping of the optical and IR LCs.

The interaction between the inner edge of the disc and the stellar magnetosphere breaks the axisymmetry of the disc material accreted towards the star. The structures formed produce changes in the optical and IR LCs. In the former, for highly inclined discs, the main mechanism is the occultation of a fraction of the stellar surface by the inner disc structures. For any inclination, the accretion pattern changes the distribution of spots leading to variations of the flux. In addition, the waves that form beyond the innermost regions of the disc may also contribute as a source of occultation in the optical

<footnote n="1">¹Departamento de Astronomía, Universidad de Guanajuato, Callejón de Jalisco, S/N, 36023 Guanajuato, México</footnote>

<footnote n="2">²Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France</footnote>

<footnote>*E-mail: e.nagel@ugto.mx (EN); jerome.bouvier@univ-grenoble-alpes.fr (JB)</footnote>
or/and emission in the IR. The MHD simulations in Romanova et al. (2013) show a bending wave moving with the stellar rotation period as well as slow, one-armed bending waves with a longer period (see their fig. 22, where such structures are clearly identified). Both waves are located close to the star and they are the result of the interaction of the inner disc, with a misaligned stellar magnetic field. Another instability able to shape this region is the Rossby wave instability (RWI), that occurs in the non-keplerian magnetized zone of a disc, where the rotational angular velocity increases with radius (Lovelace & Romanova 2014). Due to the difference in angular velocity between the star and the disc and to the presence of a stellar dipolar magnetic field, there is a transition ring, where the previous condition is fulfilled. The RWI is triggered there by an axisymmetric concentration of material, producing asymmetric modes of oscillation very close to the star, candidates to be occulting structures shaping the dips in the optical LC.

The consistent analysis of LCs in both wavelengths ranges requires simultaneous observations. McGinnis et al. (2015) presented *CoRoT* and *Spitzer* LCs observed simultaneously for YSOs in the star forming region NGC 2264. They focus on objects showing periodic dips in the *CoRoT* LC which can be interpreted as stellar occultation by a persistent warp at the inner disc edge. The behaviour is analogous to the prototypical object AA Tau (Bouvier et al. 1999), which named this category as AA Tau-like or dipper objects. A parametric study of the effect of inner disc warps and hot spots in the optical and IR LCs of YSOs is presented in Kesseli et al. (2016). The hot spots are formed when the material flowing along the *B*<sub>*</sub> lines impacts the surface. A Monte Carlo radiation transfer code (Whitney et al. 2013) is used to obtain the flux of a disc and envelope, heated by the star and the spots. The main model parameters in Kesseli et al. (2016) are the number, latitude and fractional area of the hot spots, accretion rate, inner disc radius, and the exponent of the disc warp shape: the latter shapes the occulting structures at the inner disc region. For the dipper stars, the typical optical amplitudes in the sample analysed in Stauffer et al. (2015) and McGinnis et al. (2015) are ~<0.6 mag, which is consistent with the amplitudes <0.7 mag, obtained in Kesseli et al. (2016).

We choose Mon-56, Mon-498, and Mon-1131 as adequate objects to model their IR LCs using the contribution of a warp formed at the inner edge of the disc, which channels material towards the stellar surface. These objects belong to the narrow flux dips category analysed by Stauffer et al. (2015), who identify a set of members in NGC 2264 with *CoRoT* LCs exhibiting short-duration periodic flux dips, caused by clumps of material located at or near the inner disc wall. These objects are also part of the list of AA Tau-like candidates, studied by McGinnis et al. (2015). Both studies agree that the main component required to explain the *CoRoT* LC are periodic extinction events occurring at a period consistent with the stellar rotation period (*P*<sub>*</sub>). The structures responsible for these events must therefore be located at or close to the corotation radius (*R*<sub>co</sub>). We assume that the magnetospheric radius (*R*<sub>mag</sub>) and *R*<sub>co</sub> are similar, such that probably these structures result from the interaction between the accreting material in the disc and the stellar magnetosphere. *R*<sub>mag</sub> is located where the ram pressure of the disc material slowly accreted towards the star equals the pressure associated to *B*<sub>*</sub> (Koenigl 1991). This physical mechanism relates *B*<sub>*</sub> with the mass accretion rate (M&#775;). McGinnis et al. (2015) and Roggero et al. (2021) among others suggest that the occulting structures can be the dusty base of the magnetospheric streams named in this context a warp. This structure can extend to a higher distance above the disc mid-plane when the dust sublimation time-scale is similar to the free-fall time. This scenario is analysed in Nagel & Bouvier (2020) and results in the

formation of dusty surfaces reaching large heights, and covering a large region in the magnetosphere.

McGinnis et al. (2015) interpreted the main periodic features in the *CoRoT* LCs associated to AA Tau-like behaviour, using a warp formed at the base of the stream that moves towards one of the magnetic poles in the inclined dipolar configuration. Their assumption of a simple magnetic structure to study the effect of accretion on the shaping of LCs is justified by the interpretation of spectroscopic observations of YSOs, with a stellar magnetic field model consisting of dipole and octupole components (Gregory & Donati 2011). McGinnis et al. (2015) and Roggero et al. (2021) mention that the interpretation for specific objects assumes that the system is in the unstable accretion regime where the accretion occurs along many *B*<sub>*</sub> lines. Occulting events caused by these secondary structures produce additional signatures in the LC.

## 2 LIGHT CURVE MODELLING

The physical components required to interpret optical and IR LCs are the stellar spots, the warp, the accreting streams, and the waves formed in the disc. We can physically connect all the components in the same framework, assuming that they can be explained by the same mechanism. The base of the warp starts at the inner edge of the disc and finish at a certain height. For some objects this vertically thin structure is enough to explain the amplitude of the LCs. For the modelling of other objects, an occulting structure with a larger height is required; a possibility is the dusty stream formed with material following the *B*<sub>*</sub> lines towards the magnetic pole, at a high latitude. For most YSOs, a typical observation covers a few periods observed in the optical LC. Then we assume that the contribution to the variability in the LCs is associated to structures showing periods equal or a little longer than *P*<sub>*</sub>. The region where the disc material rotates on this time-scale is located at or beyond the magnetospheric radius, and is a main contributor in the IR. According to Romanova et al. (2013), the interaction between the magnetosphere and the disc has a strong effect around *R*<sub>mag</sub>. Inside this location the inner signature is the disc warp and the streams, while outside the imprint is a bending wave. In Romanova et al. (2013), the waves are formed with rotational periods equal to or larger than *P*<sub>*</sub>. An example of the first kind is the bending wave located inside the outer vertical resonance at *R*<sub>ovr</sub> = 4<sup>1/3</sup>*R*<sub>mag</sub>. According to Romanova et al. (2013), the height of the bending wave is similar to the height of the warp. However, since the former is located at a larger radius, this structure may not occult the central star, but merely contribute to the near-IR flux. We therefore include such structures in the LC modelling.

The disc IR emission comes from two sources: one is internal heating and the other is stellar irradiation. Both sources produce a flux emitted from every section of the entire disc surface. In a quiet disc, the flux can be assumed as a non-periodic and low variability contribution. However a source of periodic variability is present when a structure, such as a wave, forms above the mean disc scale height, periodically showing to the observer sides that are directly illuminated by the star, and sides that are not. The emission is larger/lower when the observer looks at the former/latter, resulting in a periodic signal with period *P*<sub>*</sub>. According to Romanova et al. (2013), the main part of the bending wave is located at an azimuth opposite to that of the warp, such that the largest contribution of the warp occultation coincides with the largest flux contribution from the wave. For suitable conditions, one thus simultaneously observes a dip in the optical LC and a peak in the IR LC. This behaviour is clearly seen in the LC of Mon-56 at BJD 55913 in fig. 22 of
McGinnis et al. (2015), where the dip in the CoRoT LC coincides with the peak of the Spitzer LC. Following this interpretation, the Spitzer LC can be modelled as two out of phase (Φ), equal period signals; one at Φ = 180° ahead of the other, resulting in a model restricted to contributions coming from material very close to the star. A shift in the sequence of dips can be interpreted as the presence of a longer period signal that may account for longer flux variations, sometimes observed in the LC. We therefore allow for longer period components in the modelling of the LC. Finally, for high inclination systems, we also include the warp occultation.

We decided to restrict the interpretation to the mechanisms producing the main features in the LCs. The main feature of the CoRoT LC is the periodic dip that we model with a warp. According to the synthetic LCs in the models by Kesseli et al. (2016), the effect of the spots can be as large as the contribution to the dip caused by the warp. Our hypothesis is that the main accretion spot is linked to the stream associated to the warp producing the periodic signature in the optical LC. Thus, any flux contribution from the spot occurs in the lapse of time associated to the dip; at a different phase range, we neglect the contribution from a spot and around the dip, any spot contribution will produce a small change in the dip depth. Thus, we assume that the main spot is totally or partially hidden by the stream that occults a fraction of the stellar surface, where the latter effect is able to explain the LC. Depending on the object, the Spitzer LC is more or less similar to the CoRoT LC. We therefore assume that there is a contribution evolving with the period P*, which is associated with occultation and emission of the warp, as the inner disc edge is heated by the star (Nagel et al. 2013). In addition, we consider emission from a bending wave at or beyond Rmag (Romanova et al. 2013). The complete description of the time-dependent behaviour of the bending wave is beyond the scope of this work, but from the analysis of the MHD simulations by Romanova et al. (2013) and Kulkarni & Romanova (2008), we know that a stationary structure located between Rmag and Rovr is expected. The bending wave is therefore modelled as a periodic signal coming from a main structure beyond Rmag; its motion around the star sets a period for the contribution to the IR LC. If longer term flux changes are observed in the Spitzer LC compared to the synthetic LC computed with a warp and a bending wave, we additionally assume the presence of secondary waves located at larger distances in the inner disc. The details of this analysis are provided in Sections 3.1, 3.2, and 3.3.

### 3 MODELLING THE DIPPERS

The modelling of the CoRoT LC specifies the contribution of the partially occulted stellar surface. Through the modelling of the Spitzer LC, we quantify the flux contribution for all the components. The flux changes according to the occultation of the stellar surface by structures evolving at the stellar rotational period, the emission of the warp at inner disc edge directly heated by the star, and the emission of the bending wave and any other longer period structure in the inner disc. The warp used in McGinnis et al. (2015) has a large angular extension as required to explain the full width of the dips occurring every rotation cycle of the star. For all the objects in the sample, the amplitude of the CoRoT LC is explained by stellar occultations due to a structure rotating at the period P*. Here, we assume that for the high-inclination objects (Mon-56, i = 77° and Mon-498, i = 76°) the dips are due to warp occultations, and for the low inclination object (Mon-1131 and i = 27°), the occulting structure is the dusty magnetospheric streams modelled as in Nagel & Bouvier (2020).

The two main physical mechanisms used to interpret the LCs in the sample of objects studied here are the periodic dips associated

to occulting events (McGinnis et al. 2015; Stauffer et al. 2015), and periodic signals associated to a bending wave and any other longer period signals (Romanova et al. 2013). Our goal is to quantify the contribution of each of these. For the optical modelling, we use the period detected in the CoRoT LCs (McGinnis et al. 2015) to identify the number of cycles present. From the observed LCs, we extract a typical value for the dip amplitude (Δ[CoRoT]typ and Δ[Spitzer]typ for the optical and the IR LCs, respectively) to fit with the free parameters. For the occulting inner disc warp, the free parameters are the system's inclination, i, and the maximum height of the warp, h*arp,max. For the occulting stream, the free parameter is i, and smax,base is the maximum size of the grains, at the base of the stream.

In the first case, the shape of the warp is used to find the dips by occultation in the synthetic LCs, and its contribution in the IR LCs. The warp height in terms of the azimuthal angle φ is given by

$$h_{\text{warp}} = h_{\text{warp,max}} \left| \cos\left(\frac{\pi(\phi - \phi_0)}{2\phi_c}\right) \right|, \quad |\phi - \phi_0| < \phi_c, \quad (1)$$

where 2φc is the angular extension associated to the warp. Note that the maximum height of the structure is located at φ = φ0. The effect of each structure in the LC is given by the parameter φc. We fix this parameter at the value 180° in order to account for the dips width. Such a value is consistent with the models in McGinnis et al. (2015). A change of this parameter impacts the dip duration, but the depth remains unchanged. For each i and h*arp,max is varied until the amplitude of the synthetic LC coincides with the typical CoRoT amplitude. The larger the value of these parameters, the larger the depth of the feature associated to it. The parametrization of h*arp derives from the warp occultation model proposed by Bouvier et al. (1999) to account for AA Tau's LC, and was also used by Fonseca et al. (2014) to interpret the emission of V354 Mon, and by McGinnis et al. (2015) and Nagel & Bouvier (2019) for a set of LCs pertaining to samples of objects in the star forming region NGC 2264.

For the second case, the modelling requires the presence of dust at Rmag, where the grains start their journey towards the star following the B* lines. The grains survive a fraction of the journey when the sublimation (tsub) and the free-fall time-scales are similar (Nagel & Bouvier 2020). The free-fall time-scale (tff) is defined by tff = Rmag/vff, with vff = √2GM*/R*√1 - R*/Rmag, where G is the gravitational constant, M* is the stellar mass, and R* is the stellar radius. For Mon-1131, tff = 8.61 h, which is similar to tsub ~ few hours (Shu, Shang & Lee 1996; Nagel & Bouvier 2020). The evolution of the grains in the funnel flow is calculated with the sublimation rate (Xu et al. 2018) and the sublimation temperature, computed using the funnel flow density estimate corresponding to M* (Nagel & Bouvier 2020). During the approach to the star, the grain size continuously decreases until it completely disappears, such that the abundance of dust decreases as the grains move along the stream lines. As the grain decreases in size its opacity increases, resulting in two competing mechanisms in the streams controlling the optical depth towards the star.

The pairs (i, h*arp, and max) obtained by fitting the CoRoT LC are used to estimate the warp emission as in Nagel et al. (2013), using the warp shape defined in equation (1). The emission comes from a sublimation wall, where its temperature is a function of gas density, and is taken for different grain species from table 3 in Pollack (1994). The gas density is taken assuming as typical the density of a stationary configuration for a settled disc (D'Alessio et al. 1998). There are two grain size distributions: one for large grains close to the mid-plane, and another for small grains in the upper layers. The transition between these two distributions is modelled as in D'Alessio (2006). For this modelling, we assume that the composition and abundance
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Interpretation of the inner disc flux of Mon56. The disc flux represented with crosses is computed for every 0.05 increments in phase. This flux is distributed in two parts. The blue line represents the bending wave flux associated to a periodic stable signal at $P_*$ and the yellow line is the residual, representing the longer period signal associated to secondary waves.</figcaption></figure>

in the optically thin atmosphere of the optically thick warp is fixed. The dust characteristics are based on the interstellar medium (Mathis, Rumpl & Nordsieck 1977). For protoplanetary discs, the grain size distribution is given by $n(a) \propto a^{-3.5}$, where the size of the small grains located in the upper layers of the wall is $a_{min} = 0.005$ µm.

Summarizing, the time dependent modelled optical flux can be calculated as:

$$F_{\text{opt}} = F_{*}, \qquad (2)$$

where $F_*$ is the flux coming from the unocculted fraction of stellar surface. The IR modelled flux is given by:

$$F_{\text{IR}} = F_{*} + F_{w} + F_{d}, \qquad (3)$$

where $F_w$ is the flux coming from the inner disc warp, and $F_d$ is the flux associated to the inner disc containing the main bending wave evolving on a time-scale equal to $P_*$, and any structure rotating at a longer period.

The emission of the disc depends on the stellar heating of its surface and the production of heat by internal processes. For the first process, the illumination of the inner waves (the bending wave and any other wave inside the inner disc) strongly depends on the location and shape of the warp and the innermost secondary structures. For the second process, the emission of material heated by friction caused by differential rotation in the disc is relevant. A full characterization of these processes is beyond the scope of this paper, however, we can estimate the emission of a typical wave above the disc surface. For a typical temperature ($T = 1000$ K; extracted from a typical profile of a viscous disc from D'Alessio et al. 1998) and an effective emitting area of the structure ($A_{\text{eff}} = R_{\text{ad}} \times \Delta\phi \times \Delta z$), where $R_{\text{ad}} = R_{\text{mag}}$ is the typical radial location, $\Delta\phi = \Pi/4$ is the typical angular extension, and $\Delta z = 0.1 R_{\text{ad}}$ is the typical height of the structure as can be seen in fig. 2 of Romanova et al. 2013), the estimate of the disc flux for Mon-56 and Mon-498 are $F_d = 1.787 \times 10^{-13} \text{ erg s}^{-1} \text{ cm}^{-2}$ and $F_d = 1.635 \times 10^{-13} \text{ erg s}^{-1} \text{ cm}^{-2}$, using $d = 760$ pc (Dahm 2008) as the distance to NGC 2264. These values are compared for Mon-56 and Mon-498 in Sections 3.1 and 3.2.

For the modelling of the inner disc contribution which is extracted to fit the Spitzer LC, we fix a set of parameters for each object according to observational data ($M_*, R_*, T_*, i, \dot{M}, P_*$). We first

<figure><img src="image_3.png" /><figcaption>**Figure 2.** The fluxes corresponding to the fitting of the CoRoT and Spitzer LCs of Mon-56. This model corresponds to $i = 77^\circ$ and $h_{\text{warp,max}} = 0.159 R_{\text{mag}}$. The purple line corresponds to the optical stellar flux (log (flux) is shifted 0.75 for easier comparison, we include a thin purple line located as a reference to the actual level of the optical flux), the red line is associated to the IR stellar flux, the yellow line corresponds to the IR warp emission, the green line is associated to the inner disc emission and the black line is the total IR flux.</figcaption></figure>

compute the flux from the star and the warp as explained above. The height of the warp is fitted by eye to explain the amplitude of the CoRoT LC. We then compute residuals between the observed Spitzer LC and the synthetic LC, which represent the inner disc contribution. We fit these residuals with a sine curve at the stellar period, which we ascribe to a bending wave. Any longer term trends are finally assigned to secondary waves in the inner disc.

## 3.1 Model for Mon-56

The system Mon-56 is a member of the narrow flux dips category shown in Stauffer et al. (2015), consistent with a high inclination. An inner disc warp occulting a fraction of the stellar surface is a reasonable assumption to account for the CoRoT LC. We model this object using a warp parametrized as in equation (1), and estimate the flux from the inner disc as explained above. The simultaneous CoRoT and Spitzer LCs cover five rotational cycles of the system ($P_* = 5.86$ d). The inner disc flux comes from a bending wave at the stellar period plus a secondary wave at a longer period, as mentioned in Section 3. In Fig. 1, we show the sinusoidal function representing the bending wave and the remaining disc flux assigned to secondary waves. The latter flux is smaller to the former, pointing out the relevance of a bending wave as modeled here. At phase equal to 2.5 and 4.5, there are maxima in the secondary waves flux, representing a signal with a longer period. This is a reasonable scenario because in the MHD simulations of a disc interacting with the $B_*$ by Romanova et al. (2013), the inner disc shows waves moving around the star with a period longer than or equal to $P_*$.

Our main result is shown, for the model with $\dot{M} = 5.75 \times 10^{-10} \text{ M}_\odot \text{yr}^{-1}$ (Venuti et al. 2014) and $M_* = 1.16 \text{ M}_\odot$, $R_* = 1.52 \text{ R}_\odot$, and $T_{\text{eff}} = 4392$ K, from (McGinnis et al. 2015). The physical model requires a large inclination, thus we choose the largest value ($i = 77^\circ$), consistent with the estimate of McGinnis et al. (2015). The value of $i$ requires $h_{\text{warp,max}} = 0.159 R_{\text{mag}}$. For a lower $i$, $h_{\text{warp,max}}$ increases, and the modelled LC is similar. In Fig. 2, we present the flux of each of the components and in Fig. 3, the synthetic and observed optical and IR LCs. The synthetic optical LC results from the occultation caused by the warp. The synthetic IR LC
<figure><img src="image_4.png" /><figcaption>**Figure 3.** The synthetic LCs fitted to the CoRoT and Spitzer LCs of Mon-56. This model corresponds to *i* = 77° and *h*<sub>warp,max</sub> = 0.159*R*<sub>mag</sub>. The purple line corresponds to the synthetic optical LC, and the green line is associated to the synthetic IR LC. The red line represents the CoRoT LC and the black line represents the Spitzer LC. For the optical LCs a value of zero corresponds to an unocculted star. The IR LCs are shifted for clarity.</figcaption></figure>

includes three components: the emission of the occulted star, the warp emission and the inner disc emission. The amplitude of $F_d$ is $\Delta F_d \sim 5 \times 10^{-13} \text{erg s}^{-1} \text{cm}^{-2}$. We hypothesize that this flux comes from the emission of a wave above the surface of the inner disc because this value is consistent with the estimate given in Section 3 for this structure. The main result is that the contribution of the inner disc corresponds to a periodic signal with period $P_*$, which can be associated to a bending wave (Romanova et al. 2013). The values of the minima and maxima of the inner disc signal show a low-amplitude variability. This is consistent with the presence of a main stable structure periodically appearing and disappearing along the line-of-sight, the bending wave, plus a longer period structure producing the low-amplitude modulation of the minima and maxima of $F_d$.

## 3.2 Model for Mon-498

The modelling of Mon-498 follows the same steps of the modelling of Mon-56 (see Section 3.1). Both objects are members of the narrow flux dips category shown in Stauffer et al. (2015) consistent with a high inclination, which suggests that a warp occulting a fraction of the stellar surface is a reasonable assumption to model the LCs. The simultaneous CoRoT and Spitzer LCs cover seven rotational cycles of the system ($P_* = 4.28$ d).

Our main result is shown for the model with $\dot{M} = 1.0 \times 10^{-8} \text{M}_\odot \text{yr}^{-1}$ (Sousa et al. 2016) and $M_* = 1.90 \text{M}_\odot$, $R_* = 2.88 \text{R}_\odot$, and $T_{\text{eff}} = 4768$ K, from McGinnis et al. (2015). The physical model requires a large inclination, thus we choose the largest value ($i = 76°$), consistent with the estimate in McGinnis et al. (2015). The value of $i$ requires $h_{\text{warp,max}} = 0.105 R_{\text{mag}}$. In Fig. 4, we present the flux of each of the components and in Fig. 5, the synthetic and observed optical and IR LCs. The amplitude of $F_d$ is $\Delta F_d \sim 3 \times 10^{-12} \text{erg s}^{-1} \text{cm}^{-2}$. We hypothesize that this flux comes from the emission of a wave above the surface of the inner disc. However, this value is one order of magnitude larger than the estimate given in Section 3, for this structure. In order to explain this discrepancy, we may envision that: (1) the temperature of the wave is larger than what we assumed, (2) its size is larger, or (3) there are other waves contributing. The first option is probably the

<figure><img src="image_5.png" /><figcaption>**Figure 4.** The fluxes corresponding to the fitting of the CoRoT and Spitzer LCs of Mon-498. This model corresponds to *i* = 76° and $h_{\text{warp,max}} = 0.105 R_{\text{mag}}$. The purple line corresponds to the optical stellar flux (log (flux) is shifted 0.65 for easier comparison, we include a thin purple line located as a reference to the actual level of the optical flux), the red line is associated to the IR stellar flux, the yellow line corresponds to the IR warp emission, the green line is associated to the inner disc emission, and the black line is the total IR flux.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5.** The synthetic LCs fitted to the CoRoT and Spitzer LCs of Mon-498. This model corresponds to *i* = 76° and $h_{\text{warp,max}} = 0.105 R_{\text{mag}}$. The purple line corresponds to the synthetic optical LC, and the green line is associated to the synthetic IR LC. The red line represents the CoRoT LC and the black line represents the Spitzer LC. For the optical LCs a value of zero corresponds to an unocculted star. The IR LCs are shifted for clarity.</figcaption></figure>

best explanation as the luminosity of Mon-498 is five times larger than the luminosity of Mon-56, resulting in a larger contribution of the irradiated warp, as can be seen in Fig. 4 compared to Mon-56 in Fig. 2. For instance, if we increase the temperature of the bending wave to $T = 1500$ K and increase the height to $z = 0.3 R_{\text{mag}}$, which is still consistent with bending waves according to the MHD simulations by Romanova et al. (2013), then the estimated flux is $F = 1.549 \times 10^{-12} \text{erg s}^{-1} \text{cm}^{-2}$, a value near $\Delta F_d$. Note that this temperature is close to the sublimation temperature. According to Xu et al. (2018), in this case the sublimation rate of the dust grains is at its lowest value, such that the grains are in the best scenario to survive. By fitting Mon-56’s IR LC, we find that the main contribution of the inner disc corresponds to a periodic signal with period $P_*$, which can be associated to a bending wave (Romanova et al. 2013). Besides, the values of the minima and maxima of the inner disc signal show
<figure><img src="image_7.png" /><figcaption>**Figure 6.** The fluxes corresponding to the fitting of the CoRoT and Spitzer LCs of Mon-1131. This model corresponds to *i* = 27° and *s*<sub>max,base</sub> = 10 µm. The purple line corresponds to the optical stellar flux (log (flux) is shifted 0.25 for easier comparison, we include a thin purple line located as a reference to the actual level of the optical flux), the red line is associated to the IR stellar flux, the green line is associated to the inner disc emission, and the black line corresponds to the total IR flux.</figcaption></figure>

a small variability, and we conclude that another structure with a
longer period contributes as well.

3.3 Model for Mon-1131

The system Mon-1131 is a member of the narrow flux dips category shown in Stauffer et al. (2015), its modelling follows the same steps as in Section 3.1 for Mon-56. However, the object is consistent with a low inclination system such that the occultation of a fraction of the stellar surface is modelled with a dusty magnetospheric stream (Nagel & Bouvier 2020). The simultaneous CoRoT and Spitzer LCs cover six rotational cycles of the system (P<sub>*</sub> = 5.18 d).

Our main result is shown, for the model with $\dot{M} = 2.14 \times 10^{-8} \text{M}_\odot \text{yr}^{-1}$ (Venuti et al. 2014), $i = 27^\circ$ (McGinnis et al. 2015) and $s_{\text{max,base}} = 10 \mu\text{m}$ with $M_* = 0.36 \text{M}_\odot$, $R_* = 2.01 \text{R}_\odot$ and $T_{\text{eff}} = 3551$ K, from McGinnis et al. (2015). In Fig. 6, we present the flux of each of the components and in Fig. 7, the synthetic and observed optical and IR LCs. The mean value of $F_d$ is $<F_d> \sim 2 \times 10^{-12} \text{erg s}^{-1} \text{cm}^{-2}$. The contribution of a ring between $R = R_{\text{mag}}$ and $R = 1.5 R_{\text{mag}}$, inclined an angle $i$ at a temperature $T = 1000$ K, is $\sim 1.5 < F_d >$. Thus, our modelling is consistent with the flux coming from the inner disc. The contribution of all the disc sections of a low inclination system is larger compared to a high inclination system where only the structures located above the hydrostatic scale height are the ones standing out. The main result of our modelling is that the contribution of the inner disc corresponds to a periodic signal with period $P_*$, which can be associated to the bending wave plus a longer period wave (Romanova et al. 2013). This accounts for the tendency of the values of the minima and maxima of the inner disc signal to decrease between cycles 2 and 4 and increase between cycles 4 and 6. This is consistent with a stable structure periodically appearing and disappearing along the line of sight, the bending wave, completed by a wave rotating around the star with a longer period.

4 CONCLUSIONS

We have modelled the *CoRoT* and *Spitzer* LCs of three systems belonging to the narrow flux dips category shown in Stauffer et al.

<figure><img src="image_8.png" /><figcaption>**Figure 7.** The synthetic LCs fitted to the CoRoT and Spitzer LCs. This model corresponds to *i* = 27° and *s*<sub>max,base</sub> = 10 µm. The purple line corresponds to the synthetic optical LC, and the green line is associated to the synthetic IR LC. The red line represents the CoRoT LC and the black line represents the Spitzer LC of Mon-1131. For the optical LCs a value of zero corresponds to an unocculted star. The IR LCs are shifted for clarity.</figcaption></figure>

(2015). Two of the objects, Mon-56 and Mon-498, are seen at high inclination, thus an occulting warp at $R_{\text{mag}}$ is relevant to explain the CoRoT LCs. The third one, Mon-1131, is seen at low inclination and its CoRoT LC is explained by the occultation of the central star by dusty magnetospheric streams (Nagel & Bouvier 2020). According to the estimate of the free-fall and sublimation time-scales, some of the dust survives along most of the stream.

From the modelling of the near-IR *Spitzer* LC modelling, we conclude that the emission from the warp is lower than the stellar flux in Mon-56 and Mon-498. In contrast, for all three objects investigated here, the inner disc emission is of order of or larger than the stellar flux. The inner disc contribution shows a variability with period $P_*$, which is 180° out of phase of the warp occultation. This is consistent with a contribution of the bending wave inside $R_{\text{ovr}}$ as suggested by MHD simulations at Romanova et al. (2013). In all three objects, we also detect a disc contribution at a period longer than $P_*$, presumably caused by a secondary wave also seen in MHD simulations, as a slow, one-armed bending wave located in the innermost region of the disc and resulting from the interaction of the inner disc, with the magnetized rotating star (Romanova et al. 2013).

Hence, the common analysis of simultaneous optical and near-IR LCs allows us to to identify the contribution of structures affecting both wavelengths: in the optical through occultation and in the IR, through occultation and emission. The CoRoT LC fixes the number of main periodic structures, shaping the LC, while the *Spitzer* LC helps to quantify the contribution of waves in the inner disc. For systems showing periodic optical LCs with a period consistent with $P_*$, the identification of a bending wave is important to characterize how strong is the influence of $B_*$ in the disc. Note that the MHD simulations in Romanova et al. (2013), show that this wave is a signature of the interaction in a stable regime. It is reasonable that using this method, this structure can be successfully found in periodic dipper systems.

ACKNOWLEDGEMENTS

E.N acknowledges the support of the University of Guanajuato where
this article was done. J.B. acknowledges funding from the European
Research Council (ERC) under the European Union's Horizon 2020
research and innovation programme (grant agreement no. 742095; SPIDI: Star-Planets-Inner Disc-Interactions, http://www.spidi-eu.org).

## DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the corresponding author.

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