1 Introduction

Stellar activity in late-type main sequence stars induces photometric modulations and apparent radial velocity variations that may hamper the detection of Earth-like planets and the measurement of their transit parameters [1], mass and atmospheric properties. A tacit assumption when modelling light and radial velocity curves is that any temporal variation in the calibrated signal from the (unresolved) host star/exoplanet can be attributed to changes in the (small) signal from the exoplanet, and is not a result of variation in either the stellar signal or instrument response.

Late-type stars (i.e., late-F, G, K and M spectral types) are known to be variable to some extent. The effect of stellar activity in the photosphere is seen in the form of spots and faculae, whose relationship with stellar parameters such as mass and age is not well understood yet. The signal of these activity patterns is modulated by the stellar rotation period, usually producing a clear peak in the flux power spectrum of the star. The evolution of active regions, characterized by their lifetime and typical size or filling factor, has also been the subject of study of several recent works involving different methods for modelling data from spotted stars [3, 4]. Also other phenomena exist which may induce periodic signals in photometric and radial velocity data. Pulsations, granulation, long term evolution of active regions and magnetic cycles may limit the capabilities of some planet searches to success and need to be subtracted or accounted for the design of searching strategies [5–7].

The most up-to-date and comprehensive information on stellar variability comes from the studies with Kepler (stability of 200/40 ppm in a 6.5 h period on a 11/7 mag (V) star) that are based on the observation and analysis of 150,000 stars taken from the first Kepler data release [8]. (RD6) have found that 80 % of M dwarfs have dispersion less than 500 ppm over a period of 12 h, while G dwarfs are the most stable group down to 40 ppm. Kepler operates in the visible (430–890 nm) where stellar photometric variability is a factor of more than 2 higher than in the near and thermal IR (the “sweet spot” of EChO) due to increasing contrast between spots and the stellar photosphere with decreasing wavelength. Timescales for stellar activity are mostly very different from those associated with single transit observations (a few hours) and so removal of this spectral variability is possible. As a case in point, photometric modulations in the host of CoRoT-7 b are of the order of 2 % [9] and yet a transit with a depth of 0.03 % was identified [10]. Analyses of observations from Kepler have yielded comparable results.

The effects of stellar activity on EChO are very different in the case of primary (transit) and occultation observations. Alterations in the spot distribution across the stellar surface can modify the transit depth (because of the changing ratio of photosphere and spotted areas on the face of the star) when multiple transit observations are considered, potentially giving rise to spurious planetary radius variations. Correction of this effect requires the use of very quiet stars or precise modelling of the stellar surface using external constraints. The situation is much simpler for occultations, where the planetary emission follows directly from the depth measurement. In this case, only activity-induced variations on the timescale of the duration of the occultation need to be corrected for to ensure that the proper stellar flux baseline is used. In the particular case of EChO, photometric monitoring in the visible will aid in the correction of activity effects in the near and thermal IR.

In this document we provide details on a tool to simulate the effects of stellar activity and we present some methodologies to minimize jitter produced by spots on multiband photometric time series. The presented data and results in this work are focused on the framework of EChO. The methodologies we present are based on a sophisticated approach to simulate time series data for spotted stars, which is presented in Section 2. A method for full light curve scaling in the IR from visible observations is presented in Section 3, while an analysis on transit depth correlations between visible and IR observations is presented in Section 4.

2 Simulation of the photosphere of active stars

A methodology to simulate spectra from the spotted photosphere of a rotating star has been developed. We use atmosphere models for low mass stars to generate synthetic spectra for the stellar surface, spots and faculae. The spectrum of the entire visible hemisphere of the star is obtained by summing the contribution of a grid of small surface elements and by considering their individual Doppler shift, area of the surface element, projection angle and limb darkening. Using such a simulator, time series spectra can be obtained covering the rotation period of the star or longer. By convolving these with the known bandpass for a specific instrument, they can be used to study the chromatic effects of spots and faculae on photometric modulation and radial velocity jitter.

Among the most recent model atmosphere grids, we use spectra from the BT-Settl database [11] generated with the Phoenix code in order to reproduce the spectral signal for every element in the photosphere. These models include revised solar oxygen abundances and cloud model, which allow reproducing the photometric and spectroscopic properties also in very low mass stars. In our simulations of spotted stars, LTE models are used both for the quiet photosphere and the active regions (spots and faculae). Non-LTE models from [12] are to be considered in order to model faculae in future versions of the code, as they have shown to better reproduce the spectral irradiance of active features over all the magnetic cycle in the case of the Sun [13]. BT-Settl models are available at resolutions defined by a sampling ≥0.05 Å, for 2600 K < T eff < 70000 K, and several values of α enhancement and metallicity. An example for a solar-type star is shown in Fig. 1.

Fig. 1
figure 1

Solar-like star spectrum for T eff=5770 K and \(\log g=4.5\) (black) generated with BT-Settl models, compared to a spectrum of a spot for the same star, assuming ΔT spot=350K

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As said, the stellar photosphere is divided into a grid of surface elements. The spatial resolution of the grid is a parameter, and 1∘×1∘ size elements have been tested to correctly reproduce most active region configurations and transiting planet effects up to ∼10−6 photometric precision. Three models with different temperatures are considered in order to build the stellar spectrum: quiet photosphere, spots and faculae. The active regions map is specified as a list of coordinates, radii and lifetimes describing each active region. Individual sizes are considered, but the same temperature contrast for both spots and faculae is assumed for all the active regions, as well as the same ratio between the area of the faculae and the area the spots (Q). Our model does not distinguish between umbra and penumbra, but adopts a mean spot contrast at an intermediate temperature, as adopted by previous approaches [14, 15]. Active region evolution (i.e., changes in size over time) is modeled as a lineal rise period followed by a constant size time and a final decay. The typical lifetime of active regions and the rising/decaying rates are introduced as parameters in the simulation. There seems to be no correlation between spots temperatures and sizes [16, 17] and activity cycles [18] in the case of the Sun. Therefore, the temperature contrast of the spots is considered to be constant in all cases (and correlated to the spectral type of the star [21]) and active region evolution only introduces changes in the filling factor and the spatial distribution of the spots.

Each active region n is modeled as a circular spot of radius R S n surrounded by a facula with a coronal shape, having an external radius \(R_{Fn} = \sqrt {Q+1} \cdot R_{Sn}\) (see Fig. 2). This has been adopted in previous approaches and succesfully reproduced spot maps for the Sun [3] and for several low mass stars with high precission light curves.

Fig. 2
figure 2

Projected map for an arbitrarily generated distribution of active regions. The quiet photosphere is represented in grey, and each active region is modeled as a cold spot (black) and a hotter surrounding area to account for faculae (white)

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Notice that in our simulations we consider each active region as a single and large spot surrounded by faculae, as this is a more simple model while it produces similar effects on photometry than considering large groups of small spots. Active regions are equally distributed in longitudes along the equator of the star, and concentrated near to belts in the northern and southern hemispheres. The combination of the distribution of positions and sizes, and the description of the active region evolution, both defined by the input parameters, allow us to select more or less uniform distributions of active regions over space and time and then observe the effects of activity variations on planet transit measurements at the desired timescales, as will be presented in Sections 3 and 4. The method is limited to the type of active regions that can be modeled as described. On the other hand, a homogeneous distribution of small spots with no significant variations over time would difficult the determination of the parameters that we use in order to model and correct the effects of activity. Further details and discussion on the model parameters and assumptions will be presented in Herrero et al. (in preparation).

The differential rotation of the star is modeled considering:

$$ P_{\theta}=P_{0}+k_{rot}\cdot P_{\text{Sun}} \cdot \cos^{2}\theta,\quad\quad P_{\text{Sun}}=3.1913^{\circ}/\text{day}, $$
(1)

where the equatorial rotation period P 0 and the differential rotation factor k r o t are specified as input parameters, and 𝜃 is the variable denoting the colatitude. Also, the inclination of the stellar axis towards the observer is a parameter and can vary from 0∘ to 90∘.

A transiting planet can also be introduced in the simulations. The planet is considered as a circular black disk and is treated in the same way as a spot with zero flux, but assuming a circular projection on the stellar disk. The radius of the planet has no dependence on wavelength (no atmosphere is considered). The photometric signal for the primary transit is generated from the planet size, the ephemeris and the orbit orientation (inclination and spin-orbit angles). Eccentricity is considered to be zero in all cases.

Together with the physical parameters of the star—planet system, the spectral range for the output data, and the time span and cadence of the data series can be adjusted. The program also creates a light curve by convolving each created spectrum with a filter bandpass specified from a database. Also, a filter with a flat transfer function can be specified and used to study the photometric signal at any wavelength range. Finally, the integration time, the telescope collecting area, the response of the detector and the target star magnitude need to be specified in order to apply photon noise statistics to the resulting fluxes. The EChO mission involves a telescope with an aperture of 1.2 m and a detector with a mean efficiency of 0.7. However, in the case of the current study photon noise is not introduced in the simulations, as we are only interested in the systematic component of noise at different passbands, which is the one introduced by activity effects, in order to evaluate the methodologies presented here.

The program reads the physical parameters for the three photospheric features of the modeled star (quiet photosphere, spots and faculae) and interpolates within the corresponding model atmosphere grids to generate the three synthetic spectra: f p (λ),f s (λ) and f f (λ). The rest of the stellar parameters (i.e. logg, metallicity, etc.) are not significantly variable over the stellar surface, and specifically for active regions. Therefore, a precise value is not critical for the purpose of our simulations, and the nearest values in the grid are considered instead of interpolating. Also, Kurucz (ATLAS9) spectra [19] are generated for the same parameters for the photosphere, spots and faculae, as these models provide information on the limb darkening profile function I(λ,μ)/I(λ,0).

First, the spectrum for an immaculate photosphere is obtained from the contribution of all the surface elements, taking into account the geometry of the element, its projection towards the observer, its limb darkening function and its radial velocity shift:

$$\begin{array}{@{}rcl@{}} f_{im}(\lambda) &=& \sum\limits_{k} f_{p}(\lambda + \Delta\lambda_{k},\mu_{k},a_{k}) \\ &=& \sum\limits_{k} f_{p}(\lambda + \Delta\lambda_{k}) \otimes \frac{I_{p}(\lambda,\mu_{k})}{I_{p}(\lambda,0)}\cdot a_{k} \cdot \mu_{k} \cdot \omega_{k}, \end{array} $$
(2)

where k denotes the surface element and μ k is the cosine of its projection angle given by

$$ \mu_{k}=\sin i \sin \theta_{k} \cos \phi_{k} + \cos i \cos \theta_{k}. $$
(3)

ϕ k and 𝜃 k are the longitude and colatitude coordinates of the surface element, respectively, and the factor ω k accounts for its visibility, which is given by

$$\omega_{k}=\left\{\begin{array}{rl} 1 &\text{if}~\mu_{k} \ge 0 \\ 0 &\text{if}~\mu_{k} < 0, \end{array} \right. $$

and a k is the area of the surface element, which can be computed from their size (the resolution of the surface grid, Δα) by

$$ a_{k}=2 \cdot \Delta\alpha \cdot \sin (\frac{\Delta\alpha}{2})\sin\theta_{k}. $$
(4)

Finally, Δλ are the Doppler shifts added to the wavelength vector,

$$ \Delta\lambda_{k}=-8.05\cdot\lambda\cdot\frac{1}{c}\cdot R_{\text{star}} \cdot \frac{2\pi}{P_{\mathrm{0}}}\sin i \sin\theta_{k} \sin\phi_{k}. $$
(5)

in km s −1, where the rotation period P rot (in days) is given in the input parameters and the stellar radius R star (in units of R ⊙) is calculated using its relation with logg and T eff.

The flux variations \(\Delta f^{\text {ar}}_{j}\) produced by the visible active regions are added to the contribution of the immaculate photosphere when computing the spectrum at each time step j. These are given by

$$\begin{array}{@{}rcl@{}} \Delta f^{\text{ar}}_{j}(\lambda)&=& \sum\limits_{k} [(f_{\mathrm{s}}(\lambda + \Delta\lambda_{k})-f_{\mathrm{p}}(\lambda + \Delta\lambda_{k}))\\ &\otimes& \frac{I_{\mathrm{s}}(\lambda,\mu_{kj})}{I_{\mathrm{s}}(\lambda,0)}\cdot a_{k} \cdot \mu_{kj} \cdot \omega_{k} \cdot p^{\mathrm{s}}_{kj} \\& &+ (f_{\mathrm{f}}(\lambda + \Delta\lambda_{k})-f_{\mathrm{p}}(\lambda + \Delta\lambda_{k})) \cdot c_{\mathrm{f}}(\mu_{kj}) \cdot a_{k} \cdot \mu_{kj} \cdot \omega_{k} \cdot p^{\mathrm{f}}_{kj}], \end{array} $$
(6)

where the first term will be negative in surface elements to account for the flux deficit produced by the spots and the second is the overflux introduced by the faculae. The factors \(p^{\mathrm {s}}_{kj}\) and \(p^{\mathrm {f}}_{kj}\) are the fractions of the surface element k covered by spot and facula, respectively. These amounts are calculated for every surface element considering their distance to the center of all the neighbour active regions at each observation j. There is also a time dependence of the projection of the surface elements, given by

$$ \mu_{kj}=\sin i \sin \theta_{k} \cos \left[\phi_{k} + \frac{2\pi}{P_{\theta}}(t_{j}-t_{0})\right] + \cos i \cos \theta_{k}. $$
(7)

Finally, c f (μ k ) accounts for the limb brightening of faculae, which is considered with a linear model:

$$ c_{f}(\mu_{k})=\chi_{\mathrm{f}}(1-\mu_{k}). $$
(8)

The linear coefficient χ f is computed for the given stellar parameters, fixing that an active region (considering the contribution of spots and faculae) must be ∼1.2 times brighter at the limb than at the center of the stellar disk [20, 22].

Each transiting planet is considered as a dark circular spot, with no radius dependence on wavelength (no atmosphere), crossing the stellar disk. For each time step j, the planet position is obtained from the given ephemeris and the flux deficit \(\Delta f^{\text {tr}}_{j}\) is computed from the occulted surface elements of the star:

$$ \Delta f^{\text{tr}}_{j}(\lambda) = -\sum\limits_{k} f_{\mathrm{p}}(\lambda + \Delta\lambda_{k}) \otimes \frac{I_{\mathrm{p}}(\lambda,\mu_{kj})}{I_{\mathrm{p}}(\lambda,0)}\cdot a_{k} \cdot \mu_{kj} \cdot \omega_{k} \cdot p^{\text{tr}}_{kj}, $$
(9)

where \(p^{\text {tr}}_{kj}\) is the fraction of the surface element k covered by the planet. In the case that an active region is partially occulted by the planet, the corresponding spot \(\left (p^{\mathrm {s}}_{kj}\right )\) and facula \(\left (p^{\mathrm {f}}_{kj}\right )\) fractions (see (6)) in the occulted surface elements are computed inside the planet covered fraction \(\left (p^{\text {tr}}_{kj}\right )\) instead.

Finally, the spectrum for the observation j is obtained by adding the contribution of the immaculate photosphere, the active regions and the transiting planet:

$$ f_{j}(\lambda)=f_{\text{im}}(\lambda) + \Delta f^{\text{ar}}_{j}(\lambda) + \Delta f^{\text{tr}}_{j}(\lambda). $$
(10)

The resulting time series spectra are convolved with the specified filter bandpass (see Fig. 3) in order to obtain a light curve and further study the photometric jitter produced by activity at the desired spectral ranges.

Fig. 3
figure 3

Normalized flux for a simulated spectrum for the Sun-like star in Fig. 1 star with several active regions generated from BT-Settl model spectra (black). In red, the transmission function of the Kepler filter

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3 Full light curve scaling

The methodology described in Section 2 allows us to cover any broad spectral range and generate time series. We have used the simulator including both the visible and the near-IR ranges to perform several tests for EChO data-type optimization. We can accurately analyse and compare activity photometric signal amplitudes and patterns at different wavelengths and spectral resolutions.

We have considered a number of combinations of stellar photospheres and active region parameters (size and location of spots, temperature contrast) to improve the statistical view of the results. The process was performed for 11 cases of star-planet systems randomly selected from a set of 5 stellar models (T eff=6200 K, 5850 K, 5060 K, 4060 K, 3580 K) having four different possible active region maps and a rotation period of 15 days. The rotation period is not a relevant parameter in this case, as we are analyzing low resolution data and Doppler broadening is negligible. The spot temperature contrast was scaled with temperature according to [21] and the planet parameters were fixed in all cases to R planet=0.05R star, P planet=2.54 days, b=0.2 (impact parameter), which correspond to a standard Hot Neptune when a Sun-like host is considered, and provide a ∼0.25 % transit depth. The facula temperature contrast and the facula-to-spot area ratio (Q) were fixed in all cases to ΔT fac=+100 K and Q=7.0, respectively. Simulations are carried out along a complete rotation period of the star and considering a much larger timescale for the active region evolution. Therefore, active regions are assumed to be unchanged during the time series, as our current study is mainly intended to present and correct the activity modulated effects on multiple spectral ranges, and change in the filling factor or distribution of spots will not have a significant effect on these. No photon shot noise is introduced in the simulations (see Section 2) so the only observable variations are the effects of modulations due to active regions. Narrow bandpasses equivalent to a single spectral resolution element of EChO (R=300) centered at 0.8 μm, 2.5 μm and 5.0 μm were used in order to generate light curves of the spotted rotating stars.

The approach we have investigated to correct near IR time series data for stellar activity is based on scaling the higher-amplitude optical light curve to the near-IR wavelengths and carry out a direct correction. This procedure was defined so that it could resemble a practical procedure to use with EChO data. The detailed steps are: 1) We first normalized the fluxes and calculated the standard deviation of the optical and near-IR photometry, σ(0.8 μm), σ(2.5 μm) and σ(5.0 μm); 2) ratios of these standard deviations (near-IR to visible), defined as K2.5=σ(2.5 μ m)/σ(0.8 μm) and K5.0=σ(5.0 μ m)/σ(0.8 μm), were calculated; 3) the scaling factors K2.5 and K5.0 were applied to the optical light curve; and 4) the final near-IR light curve was obtained from the ratio of the generated time series to the scaled optical light curve. Tests showed that the time modulation of the near-IR light curves could be significantly removed or corrected following this approach. The sequence above was both applied to a light curve without exoplanet transits and to a light curve including transits.

We evaluated the possible performance of the method in removing activity signals by comparing the resulting corrected light curve with a light curve including transits and corresponding to a star with an immaculate photosphere. This was carried out to check whether systematic effects inside the transits were left over when following the procedure. Indeed, depending on the distribution of starspots and the trajectory of the planet across the stellar disk, the depth of the transit could experience variations (which would mask or bias the measurement of the planetary atmosphere) both in the case of spot crossing events but also if only photosphere is occulted (e.g., [23]).

In our test we divided the resulting clean IR light curves (i.e., corrected by scaling the 0.8 μm light curve as described above) by light curves generated for an immaculate star (with no spots) for the same stellar and planet parameters. Then, data intervals of 2⋅T14 (i.e., twice the total duration of the transit) around the transit events were selected for analysis. The out-of-transit baselines at either side of the transit were used to fit a linear trend that was subsequently applied to the in-transit observations. This is analogous to the common procedure used to analyse and fit transit data. The standard deviations were computed for the in-transit sections of the complete data series (which includes 10 transits). The results of this set of simulations can be seen in Table 1 and Fig. 4. Table 1 shows the stellar parameters adopted and standard deviation caused by the spot modulation at 0.8 μm, 2.5 μm and 5.0 μm bandpasses. The last two columns list the standard deviation of the in-transit data at 2.5 μm and 5.0 μm after correcting for spot modulation using the procedure described above.

Fig. 4
figure 4

Top: Normalized light curves for three narrow passbands (R =300) centered at 0.8 μm (black), 2.5 μm (red) and 5.0 μm (green) for one of the cases generated in the sample. The standard deviations of the data series, used in order to compute the scale factors, are also indicated. Bottom: 2.5 μm (red) and 5.0 μm (green) light curves for the same star including a transiting planet, and corrected from the activity signal by using the scaled 0.8 μm light curve. The light curves for an immaculate star are shown in blue

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Table 1 Results for the simulations of 11 cases of star-planet systems randomly selected from a set of 6 stellar models and 4 different possible active region maps, and with a rotation period of 15 days
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As can be seen in Table 1, most of the cases (10) that we have analysed thus far correspond to standard stars of GKM spectral types with filling factors of 0.5 % to 0.9 %. To define the context, our Sun ranges from filling factors of ∼0 % to ∼0.2 %. So, the stellar parameters that we have simulated correspond to stars that are some 2-3 times more active than the active Sun. According to the Kepler mission results (e.g., [2]), this would be representative of the 20 % of the solar neighborhood population and thus indicate a statistical upper limit to what we can expect of the exoplanet host distribution. As can be seen from Table 1, in the first 10 cases the plain direct procedure that we have used here provides a correction to the transit data to better than 10−4 and often as good as a few parts in 10−5. Note also that, if obvious photometric outliers corresponding to spot crossing events are cleaned, the results are further improved and in all cases the residual standard deviation is below ∼1.3 ⋅10−5.

The last row of Table 1 corresponds to a rather active star with similar parameters to the exoplanet host HD 189733. The filling factor is around 5 % and in this case the standard deviation of the 0.8 μm, 2.5 μm and 5.0 μm light curves is of the order of 5–10 times greater than in the previous 10 cases. However, the results show that the scaling procedure works quite efficiently in this case as well and the standard deviations of the corrected transit events is of ∼1 ⋅10−4. The standard deviation is further reduced to ∼3 ⋅10−5 when spot crossing outliers are cleaned.

The procedure explained here constitutes a direct methodology to correct transit data that may be the simplest approach to address stellar activity effects. Even in this case, our results show that it is possible to reach precisions better than 10−4 on transit data in the near IR for most of the stellar host population. If further care is taken and outliers caused by spot crossing events are cleaned, the precision reachable approaches 10−5 or better in most instances. This is even the case of a moderately active exoplanet host such as HD 189733. The tests that we present show that the final in-transit standard deviation is a factor 2 to 5 ⋅102 smaller than the original standard deviation of the light curve at 0.8 μm. This simple rule-of-thumb can be used to assess the maximum photometric activity tolerable to attain the precision requirements of EChO with this direct scaling technique. More strongly active targets, which may exist considering that EChO will observe M-type planet hosts, are very likely to require a more sophisticated analysis methodology in order to reach precisions in the range 10−4 to 10−5.

A shortcoming of this methodology is that it requires continuous or nearly-continuous spectrophotometry to model the out-of-transit variability and calculate the scaling factors. In the case of EChO, this may occur for some transiting planets with a very large number of observations or may require supporting ground-based observations to monitor the photometric variations. Another drawback is that the transit depth measured once the activity induced variations are removed is the one that would result with the mean spots filling factor from the overall time series, but a zero point remains when applying this method. Given these restrictions we have explored another method to correct transit and eclipse data that does not require other observations than those of the events themselves.

4 VIS-IR transit depth correlations

In view of the need of continuous spectrophotometry to apply the methodology in Section 3, we have investigated a direct method of correlating activity-induced variations in the visible with those in the IR. The underlying hypothesis is that variations of the transit depth over time in the visible are solely caused by stellar activity effects (mainly modulations at the timescale of the rotation period) and not influenced by the atmosphere of the transiting planet, which in turn is known to produce a transit depth dependence on wavelength [24]. The spectral signal of the planetary atmosphere in the visible is significantly weaker than in the IR and time variations should be negligible, thus lending support to the working hypothesis. In this framework, by characterizing variations in the visible it should be possible to correct out activity-induced variations in the IR.

To simulate the performance of the method we have considered a random selection for the same combinations of stellar photospheres, active region parameters and transiting planet parameters as in Section 3. The list of cases with initial parameters and results is presented in Table 2. Note that we are now considering a wider range of filling factors than in the case of Section 3, so that the effects of both non-occulted spots and spot crossing events will be more easily studied than in the case of rather inactive stars on a short series of transit observations.

Table 2 Results for the simulations of the 10 cases randomly selected from the same parameter combinations as in Table 1, but considering a wider range of filling factors
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From the modeled light curve we have measured the transit depths, using common techniques, and calculated the variations of those depths with time. This was repeated for all the transit events (60 for each object) of all our simulated cases. The combined dataset of all transit depths was then analysed for correlations. We have found that there is a well-defined correlation between activity-induced transit depth variations in the visible (0.8 μm) and the IR (2.5 and 5.0 μm). An illustration of the transit light curves generated by the simulator and the correlation between visible and IR transit depth variations (TDV) can be seen in Fig. 5, including linear regression fits.

Fig. 5
figure 5

Left: Three of the transit light curves at 2.5 μm (red) and 5.0 μm (green) for one of the cases generated in the sample, compared with the transit light curve of an immaculate star (blue). Note the small systematic deviations and the more apparent spot crossing events. Right: Correlation of activity-induced transit depth variations (TDV) for all the studied cases and transit events in the visible (0.8 μm) and the IR (2.5 and 5.0 μm)

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The methodology for correcting EChO data for stellar activity effects using, for example, a series of measurements in the visible and an IR band can be carried out using the following expression:

$$ d^{corr}_{IR}=d_{IR}+a_{0}+a_{1}\cdot (d_{VIS}-\langle d_{VIS}\rangle ), $$
(11)

where d stands for the transit depth, and a 0 and a 1 are the coefficients of a linear fit that can be determined from simulations (shown in Fig. 5 as an example for our simulated cases). Actually, our tests show that a 0 takes negligible values and can be adopted as 0.

We tested the performance of the method by coming back to our simulated transit depth data. The linear regression takes values a 1=0.438 for the 2.5 μm data and a 1=0.368 for 5.0 μm. For each transit event, we used the general coefficients and the visible TDV to correct the IR depth for stellar activity. Then, we analysed the resulting scatter in the depth variations of the combined events for each simulated case. The results can be seen in Table 2. The cases that we have analysed represent standard stars of GKM spectral types with filling factors of 1–7 %, i.e., corresponding to stars that are ∼4–30 times more spotted than the active Sun. The case in row 1 has parameters similar to HD 189733. As can be seen from Table 2, we start with initial transit depth rms values that are generally around 10−3. Correction using the procedure described decreases such rms by factors ranging from 20 to several hundreds. Thus, the resulting rms in the transit depth variations in the IR shows that this procedure provides a correction of the transit data to a few times 10−5. For more active stars (which will be a very small subsample, anyway) the correction may not reach this high performance, but the final TDV rms is very likely not to exceed 10−4, and in any event stay fully compliant with EChO noise requirements.

Although we are significantly removing the variations induced by activity, again as in the methodology presented in Section 3 we are obtaining a transit depth measurement, \(d^{corr}_{IR}\), that depends on the mean spot filling factor over the data time series. That is to say, the results are corrected from the variability, but not from the zero point, thus providing a biased planet radius. However, the methodology is intended to correct for the spectral dependence of activity effects in order to study the atmosphere of the planet. Our approach succeeds to remove TDVs up to the EChO requirements in order to be able to study the spectral signature of the planet atmosphere when averaging the series of transits. Also notice that we assumed the temperature contrast of the spots to be unchanged during the observations. No correlations have been observed between the temperature contrast and other properties of the spots or the activity cycle for the Sun [18, 25, 26]. Large variations in the temperature of the active regions would modify their chromatic signal and hence would affect the determination of the parameter a 1.

5 Conclusions

We presented a methodology to generate simulated time series spectro-photometry for rotating spotted stars, which provides us some tools to characterize the effects of activity jitters on data as generated by the EChO mission. We used two approaches in order to correct the data by significantly removing jitter caused by active regions on light curves, especially in the case of transit data, when simultaneous observations in the visible and the IR ranges are available.

We proved that activity effects introduced in transit measurements can be removed up to 10−4 photometric precision if the IR band data are divided by the scaled data in the visible band (see Section 2). About ∼3 ⋅10−5 precisions are achieved if spot crossing events are independently removed. However, as continuous monitoring of all targets will not be available from EChO, a second approach to correct for activity jitter is presented considering only a series of transit observations. A correlation of transit depth variations between visible and IR bands is found. Thus, our methodology directly provides the coefficients to correct transit depth measurements from the effects of non-occulted spots when simultaneous multiband observations are available, as in the case of EChO. The results in Section 3 show that up to 10−5 photometric precision can be achieved when correcting transit depth variations even in significantly active stars (2–10 times more than the active Sun).