THE MASS OF CoRoT-7b

Hatzes et al. (2010) used a simple method for removing the RV signal due to activity. The method exploits the fact that the RV variations due to the transiting planet have much shorter timescales than those expected from the activity signal. Only those HARPS data that had multiple measurements taken on the same night and with good time separation between successive measurements were used. This resulted in a total of 64 measurements or about one-half of the HARPS RV data (a case of less being more). (Note that this includes an additional four nights of data inadvertently left out by Hatzes et al. 2010). Table 2 lists the starting Julian Day for the nightly measurements, the number of measurements taken on that night, the time separation between first and last nightly measurement, and the resulting orbital phase difference between the first and last observation on that night.

There are two very simple assumptions in this analysis: (1) a 0.85 day RV period is present in the data. (2) The RV zero-point offset does not vary significantly over the span of measurements for a single night.

Assumption (1) is reasonable given the clear transit signal in the data. Léger et al. (2009) carefully excluded all possible false positives and established with a high degree of confidence that a planetary companion was causing the transit event. Hatzes et al. (2010) showed that even this assumption can be relaxed as the 0.85 day period provides the best fit to the data. Later we show a periodogram analysis that also confirms that a 0.85-d period provides the best fit to the RV data.

Assumption (2) is also eminently reasonable. The rotation period is known to be 23 days and over the time span of the nightly observations (maximum ≈4 hr) the star rotates by no more than 26. The RV amplitude due to activity rotational modulation is ≈10 m s⁻¹ (Queloz et al. 2009). This means that the change in RV due to rotational modulation will amount to a maximum value of < 0.5 m s⁻¹. This is less than the expected change in the RV due to the orbital motion of CoRoT-7b which is 1–5 m s⁻¹ and with an average of about 3 m s⁻¹ on a given night. Rapid and significant changes in the spot distribution on timescales of ≈4 hr in a star with solar-like activity like CoRoT-7 would be unprecedented. For instance, measurements of the solar constant from the Solar Maximum Mission shows peak variations of 0.2% on timescales only as short as weeks (Foukal 1987), i.e., much longer than the timescales of our nightly variations or the orbital period of CoRoT-7b. Photometric variability does not necessarily translate into RV variability and vice versa, but drawing from the solar analogy we do not expect significant activity related RV variations on timescales of a few hours. In short, we expect that the RV contribution due to activity over the time span of the nightly measurements should be nearly constant.

If there are additional planets present then these might contribute to changes in the mean RV for a given night that would not subtract out fully. These, however, also make a negligible contribution. For the sake of argument let us assume that CoRoT-7c and CoRoT-7d are also present. Using the orbital solutions of Hatzes et al. (2010), we can calculate the change in RV on a given night due to the reflex motion of the star caused by these companions. This is listed in the fifth column of Table 2 under ΔV_{C7c, d}. (Note that due to the shorter period the largest effect stems from CoRoT-7c.) The average change in RV is −0.04 with an rms scatter of only 0.89 m s⁻¹. The maximum change in RV of 2 m s⁻¹ only adds a maximum possible error of ±1 m s⁻¹ to the zero-point offset. For most nights, the error is less than 0.5–1 m s⁻¹. Combined with the rotational modulation signal, the error in the nightly RV zero-point offsets (mean values) is less than the measurement error of about 2 m s⁻¹ as well as the typical change in the RV due to orbital motion. We should emphasize that it is irrelevant whether CoRoT-7c and CoRoT-7d exist or not. If they are actually caused by stellar activity then our orbital solution can be treated as a "model" for this activity that still provides a good estimate for the short-term activity RV jitter.

We stress that in this procedure we essentially do not care what the contribution of surface spots or other planets is to the RV signal. By calculating the mean RV value for a given night (in a least-squares sense) and subtracting this from the signal, we effectively remove all other contributions to the RV signal that have periods substantially longer than the orbital period of CoRoT-7b. Our approach will also remove any unknown long-term systematic instrumental errors that may be present. The stability of the HARPS spectrograph is nearly legendary with nightly drifts of the spectrograph on average being less than 0.5 m s⁻¹ (Lo Curto et al. 2010). Even if HARPS had night-to-night systematic errors, our approach should eliminate, or at least greatly minimize these. As long as the RV variations from other sources do not change significantly over a 2–4 hr time span, any RV variations that are seen in a given night can be attributed solely to that of CoRoT-7b.

An orbital solution was made using the generalized nonlinear least-squares program Gaussfit (Jefferys et al. 1988) and treating data from individual nights as independent data sets with their own velocity (γ-velocity) offset. (For convenience, we refer to this as the "model-independent" approach.) Such a technique has been used successfully to compute orbital solutions by combining multiple RV data sets taken with different instruments and using different RV measurement techniques (for example, see Hatzes et al. 2003). The period was fixed to the CoRoT-determined transit value of 0.853585 days, but the phase, amplitude, and nightly offsets were allowed to vary until the best-fit solution was found that minimized the χ². The calculated phase reproduced the transit phase to within 0.01. Varying the orbital eccentricity resulted in a small eccentricity (e = 0.077) but with large errors (±0.11). The orbit can be considered to be circular. For the final solution, we used a zero eccentricity and fixed the orbital phase to the transit value.

Figure 1 shows the nightly RV measurements after removal of the nightly zero-point offsets and phased to the CoRoT transit ephemeris. The resulting K-amplitude is 5.15 ± 0.95 m s⁻¹. Measurements taken in a single night are represented by the same symbol and one clearly sees that the change in the RV almost always follows the orbit to within the measurement error. The rms scatter of the RV data about the orbital solution (line) is 1.68 m s⁻¹ which is in excellent agreement with the median RV internal error of 1.77 m s⁻¹. This figure alone argues that the contribution of activity jitter to this RV curve is negligible.

Zoom In Zoom Out Reset image size

We also see no obvious evidence for any short-term (i.e., during the night) systematic errors in the RV data (possible long-term errors are removed by subtracting the nightly mean), otherwise we would see a scatter about the orbital solution significantly larger than the typical measurement error. There are several dozen cases where the RV measurements of stars are in the same signal-to-noise regime as CoRoT-7 and these show a residual scatter in the range of 1.5–3.5 m s⁻¹ (Naef et al. 2010; Lo Curto et al. 2010; Moutou et al. 2011), comparable to what we see in CoRoT-7.

Our Keplerian solution from the previous section established that the orbit of CoRoT-7b is circular and in phase with the transit light curve. We can thus use a linear model of the form:

$$\begin{equation} A\cos (\omega t_i) + B\sin (\omega t_i) + C_i, \end{equation} \tag{ 1 }$$

where A and B determine the amplitude and phase of the sine function fitted to the RV periodic modulation, and C_i ∈ {C₁, ..., C₂₇} are the different shifts for each of the 27 nights with multiple measurements (each C_i is assumed to be constant for each night), ω = 2π/P for a given period P and t_i is the time of each measurement. Presenting the model in the form of Equation (1) enabled us to apply a linear model to the data as the phase does not appear explicitly thus assuring a robustness of the best solution. The resulting RV amplitude from the linear model was 5.25 ±0.84 m s⁻¹, entirely consistent with the Gaussfit result. We shall use this value as our "best-fit" K-amplitude. In order to assess the significance of the detected periodicity, we applied the linear model to the RV measurements using a range of different frequencies, to see weather our analysis detected any modulation with another periodicity. The periodogram of derived 1/χ² as a function of frequency in the range (0,2] day⁻¹ (Figure 2) shows a clear global maximum near CoRoT-7b's period, which was found through photometry by Léger et al. (2009).

Zoom In Zoom Out Reset image size

We also performed a bootstrap test to see if random data can produce the same modulation. In the test, all RVs used in the analysis were randomly shuffled, leaving the observing times the same. We performed 100,000 runs, each contained 1000 frequencies at the same range taken in the periodogram of Figure 2. Out of 10⁸ linear fits, the highest 1/χ² value was 0.24, while the original data yielded a maximum 1/χ² of 1.2, near the Corot-7b period. The results are plotted in Figure 3, which shows a histogram of the highest 1/χ² values for all 100,000 runs. We conclude that the RV modulation with the photometric period was detected with a significance higher than 10⁻⁵.

Zoom In Zoom Out Reset image size

The finite differences of the nightly RV measurements can also be used to estimate a K-amplitude that is largely insensitive to the contribution of the activity. Differences (derivative) should remove the underlying activity signal assumed to be constant for a given night so that any changes in the RV can be attributed solely to orbital motion. The fitting to the RV data done in the previous sections confirms that the orbit is circular and in phase with the CoRoT ephemeris. The RV curve can thus be described by the function: V = −Ksin (2πϕ) + C_i, where K is the RV amplitude, ϕ the orbital phase, C_i is the nightly zero point, and the "minus" sign ensures that the RV curve has the proper phase with respect to the mid-transit time. Differentiating with respect to ϕ results in dV/dϕ = −2πKcos(2πϕ). So, a plot of the derivative (differences) of the RV values versus cos(ϕ) should have a linear relationship whose slope is proportional to the K-amplitude.

Figure 4 shows the normalized nightly RV derivatives (dV/dϕ/2π) as a function of −cos(ϕ). There is considerable scatter since we are dealing with differences, but there is a clear correlation. The correlation coefficient is r = 0.59 with a probability of 6 × 10⁻⁵ that the data are uncorrelated. A least-squares fit to the derivatives yields a K-amplitude of 5.02 ± 1.25 m s⁻¹ a value entirely consistent with the previous approaches.

Zoom In Zoom Out Reset image size

Since the derivative of a circular orbit is also a trigonometric function one should also be able to fit a sine function (with the proper phase) to the finite differences. Doing this resulted in a K-amplitude of 6.09 ± 1.42 m s⁻¹. In summary, the finite differ- ence also demonstrates the presence of nightly variations consistent with a period of 0.85 day and a K-amplitude of ≈5 m s⁻¹.

We assessed the robustness of our K-amplitude determination through a series of tests. First, we investigated how the calculated K-amplitude depended on the amount of data that was used. The top of Figure 5 shows the K-amplitude as a function of number of nights of data used in the solution. After five nights of data, the K-amplitude converges to within the errors to the nominal value using all the data (horizontal dashed line). The final two points at the right show the K-amplitude determined using those nights with only two measurements (K = 4.00 ± 1.56 m s⁻¹) and nights with three measurements (K = 5.87 ± 1.21 m s⁻¹). Both agree to the nominal value within the errors.

Zoom In Zoom Out Reset image size

We also performed Monte Carlo simulations to confirm the error on the K-amplitude, but more importantly to assess how well we could recover a known signal in the RV data. Given the relative sparse sampling of the sine curve on each night, there was some concern that when fitting a fixed period to the data the procedure may introduce variations with that period into the data due primarily to the freedom in varying the nightly offsets. A synthetic sine wave was generated with an amplitude of 5 m s⁻¹ and having the same period and phase of CoRoT-7b. The fake data were sampled in the same manner as the real data and random noise at a level of 2 m⁻¹ were added. To mimic the activity signal, random noise with an rms of 10 m s⁻¹ was added to the fake RVs for each night. The data were then processed in the same manner as the real data. One hundred such simulations yielded a mean amplitude of 5.00 ± 0.94 m s⁻¹ and in all cases the nightly "activity" signal was recovered to within the measurement error. The error in the K-amplitude from the simulations is entirely consistent with the measured value and suggests that the errors in the RV measurements are nearly Gaussian. The procedure also is able to recover the input amplitude over a wide range of input K-amplitudes (0–10 m s⁻¹) as shown in lower panel of Figure 5. Each point in this figure represents an average amplitude from 20 simulated data sets.

The previous simulations assumed a random component to the underlying activity signal. However, this component is most likely periodic and we should worry about the activity related RV variations with a frequency of four times the rotational frequency, 4ν_rot since this may correlate with an 0.85 day period through the 1 day alias (see below). This might introduce a significant contribution to the recovered K-amplitude. As a final test, we used a sine wave with a period of one-fourth the rotation period to provide the value for the constant RV offset due to activity. A discrete Fourier transform (DFT) of the raw HARPS RV data yields an amplitude of 4.6 m s⁻¹ near 4ν_rot. Although this is mostly due to the 1 day alias of the orbital frequency of CoRoT-7b, we used this as a "worst case scenario" for the contribution to the activity signal due to 4ν_rot. An input amplitude of 5 m s⁻¹ was then used for planet K-amplitude and the phase of the underlying "activity" sine wave was varied over the full range (0–2π). The recovered K-amplitude ranged from 4.6 to 5.6 m s⁻¹, i.e., within the error of our nominal value. We conclude that we can reliably recover the K-amplitude of CoRoT-7b even in the presence of activity that is correlated with the rotational period.

The nightly RV offsets calculated by our procedure should have some relationship with the underlying activity signal which should be dominated by the rotation period. To check this, we performed a Scargle periodogram (Scargle 1982) on the calculated nightly offsets. This periodogram, shown in Figure 6, has its highest peak at a frequency of ν = 0.043 ± 0.002 c d⁻¹ which is consistent with the rotational frequency for the star of 0.0423 ± 0.006 c d⁻¹ from Lanza et al. (2010). According to Scargle (1982), the probability that noise will produce this peak exactly at the rotational frequency is about 5 × 10⁻⁴. This was confirmed using a bootstrap randomization procedure. The RV data were randomly shuffled 200,000 times, keeping the times fixed and the maximum power in the random data periodogram was examined. Over the narrow frequency range 0.035–0.055 c d⁻¹ centered on the rotational frequency, the false alarm probability that noise is causing the observed peak was 7 × 10⁻⁵. The derived nightly offsets thus have some relationship to a known phenomenon associated with the star (i.e., rotation)—these are not arbitrary. The RV amplitude of this peak is ≈10 m s⁻¹, consistent with the RV amplitude due to rotational modulation using the entire HARPS data set (Hatzes et al. 2010).

Zoom In Zoom Out Reset image size

Queloz et al. (2009) used a method similar to that of this paper to study the possible variations of the CCF bisector induced by the RVs variations due to the CoRoT-7b signal. For each of the observing nights with two or three measurements per night, they corrected the RVs from their average during the night over the two or three measurements. They did the same correction on the bisector span by their nightly average. These "differential" bisector spans and RVs can show short-period variations, on a timescale of about 5 hr which is the typical longest time offset between data on nights with multiple observations. This represents 25% of the CoRoT-7b orbit.

The nightly variations of the bisector span as a function of the nightly variations of the RVs are plotted in Figure 7. It shows peak-to-peak variations on the order of 20 m s⁻¹ for the bisector spans and 10 m s⁻¹ for the RVs. The larger dispersion of the bisectors agrees with their error bars that are twice as large as than those for the RVs. The bisectors show a hint for correlation with the RVs. With the hypothesis of a linear correlation, the data show a positive slope of 0.45 ± 0.22 that is plotted in Figure 7. The Spearman's rank test indicates a 2σ deviation from the null correlation hypothesis. A bootstrap with 30,000 pools mixing RV and bisectors shows a slope distribution centered on zero, with only 1.9% probability to find a slope of 0.45 or larger. It also shows a 2.1% probability to find by chance a deviation from the null correlation hypothesis greater than 2σ with the Spearman's test.

Zoom In Zoom Out Reset image size

Thus, the data show a 2σ detection of a positive correlation between the differential bisector spans and RVs, on a night-per-night basis. Such correlation could be the signature of a blend scenario (e.g., Santos et al. 2002), where a star in the background of the main target hosts a deep transit which seems shallower as it is diluted by the constant, brighter flux of the main target. However, blend simulations presented by Queloz et al. (2009) have shown that the slope of the bisector versus the RV should be steeper than 2 in all cases, whereas 0.45 ± 0.22 is found here. Thus, the possible correlation between bisector spans and RVs apparently cannot be explained with a blend scenario. In addition, such a blend scenario would have to imply that the background star hosts a deep transiting companion while the main star, by chance, hosts at least one planet (CoRoT-7c) as detected in RVs. A planetary system is more likely with the two companions orbiting the same stars.

To investigate further any possible bisector variations, the same analysis using Gaussfit that was applied to the RV data was also applied to the bisector data. Namely, the nightly data were treated as independent data sets and a sine wave was fit to the data keeping the period fixed at 0.85 days and allowing the nightly means to float. If we find strong sinusoidal variations with a 0.85 day period it would imply that either that activity is contributing significantly to the RV signal, or that our procedure is introducing RV variations with the planet orbital period. The top of the Figure 8 shows the residual bisector variations after subtracting the calculated nightly mean and phased to the CoRoT-7b period. There is no compelling evidence for sinusoidal variations. A sine fit to the data yields an amplitude of 2.8 ± 1.1 m s⁻¹. The only hint of variability is driven by a cluster of only four points at phase ≈0.1. Eliminating these points in a much lower amplitude of 1.6 ± 0.9 m s⁻¹. Most of the bisector span data between phases 0.2 and 1.0 have an rms scatter of ±1σ as shown by the dashed lines. For now any possible bisector correlation or variations remain unexplained, but one should keep in mind that these variations with the present data set do not seem to be significant. Even though there are often large nightly variations, these are all within the rms scatter of about 4.5 m s⁻¹. (Compared to the ≈10 m s⁻¹ rms scatter for the bisector variations of Kepler-10b, see Figure 7 of Batalha et al. 2011.) This also argues against any possible variations being due to a blend scenario.

Zoom In Zoom Out Reset image size

We also applied this method to the FWHM measurements. This is shown in the bottom panel of Figure 8 where we plot the FWHM residuals phased to the 0.85 day period. The amplitude of any variations is 3.8 ± 3.2 m s⁻¹. There is no strong evidence for variability in the FWHM with a 0.85 day period.

There are two possibilities for the RV curve shown in Figure 1: either it is dominated by the reflex motion of the star due to CoRoT-7b, or the signal due to activity is producing a significant contribution to the amplitude. Pont et al. (2011) derived a significantly lower K-amplitude of 1.6 ± 1.3 m s⁻¹ after employing their model for the activity on CoRoT-7. If this value is correct, then this implies that a significant contribution (about 3.5 m s⁻¹) to the RV amplitude comes from the contribution of activity.

The striking feature about the RV curve (Figure 1) is that it does not deviate from a circular orbit (i.e., pure sine curve) to within the errors using the period and phase of the photometric transit. Naively, we would expect an uncorrected activity signal to distort significantly the RV curve—it should not be so "clean." We believe that the stellar activity actually contributes little to the observed RV amplitude.

For activity to contribute significantly to the RV curve, it would have to have variations that would mimic the 0.85 day period and be perfectly in phase with the CoRoT-7b lightcurve ephemeris, otherwise we could not fit a circular orbit to the data. We could concoct a spot distribution that evolves with just the right timescales and phase with respect to the 0.85 day period, but that would be too ad hoc and lacking in a reasonable physical interpretation. The simplest and most logical way to produce a 0.85 day variation in the RV data, and one that has a stronger physical basis, is with the 1 day alias of P_rot/4 (ν = 4ν_rot + 1) which is close to the CoRoT-7b orbital frequency. We emphasize that Figure 8 of Hatzes et al. (2010) and the periodogram in Figure 2 demonstrate that the true period of 0.85 day is favored over the alias period and that it provides a better fit to the data. However, for the sake of argument let us assume that a large part of the RV amplitude in Figure 1 is indeed due to an alias of 4ν_rot rotational harmonic. We argue that this cannot be the case for two strong reasons.

First, there is little evidence for 4ν_rot actually being present in the activity indicators. Assuming the Pont et al. K-amplitude of 1.6 m s⁻¹ is indeed correct, then ≈3.5 m s⁻¹ of the RV amplitude has to be accounted for by activity. The RV amplitude associated with the rotational frequency is 8.7 m s⁻¹. This means that the fourth harmonic (ν_rot) should have an amplitude ≈0.4 of the rotational peak. We find, however, weak evidence for significant power at 4ν_rot. Figure 9 shows the DFT amplitude spectra of the FWHM, the bisectors, and the Ca ii emission. The vertical line marks the location of 4ν_rot which is near the 1 day alias of the orbital frequency of CoRoT-7b. Only the FWHM and Ca ii show a bit of power near this rotational harmonic. The FWHM shows a ratio of the fourth harmonic to rotational frequency of only 0.2. Ca ii does show a peak at the third harmonic with the proper amplitude (≈0.4). However, the DFT of both the FWHM and bisector span are quite noisy with other peaks close to the fourth harmonic having a larger amplitude. Pont et al. (2011) argue that the FWHM can be used to reconstruct the brightness variations to 0.1% or better. If this is true, and since 4ν_rot has an amplitude 20% of the rotational peak, then the RV variations due to the fourth harmonic should be about 1.7 m s⁻¹, or comparable to the measurement error.

Zoom In Zoom Out Reset image size

Second, and just as importantly, we would have to place very stringent constraints on the spot distribution and its evolution that are highly unlikely:

• 1.  
  Physically, the only way to produce a strong presence of P_rot/4 in the RV rotational modulation is to have four spot groups equally spaced in longitude by 90°. Each spot group would produce its own sinusoidal variation and all four RV curves from the individual spot groups would add together to contribute to the observed 0.85 day RV variations (in this case, the 1 day alias of P_rot/4.)
• 2.  
  For activity to reproduce the CoRoT transit phase and to be able to add or subtract to the RV curve in the proper phase, one spot group must be located at transit phase zero. Otherwise, the activity signal would introduce significant distortions to the RV curve. We cannot dismiss that the close proximity of the planet actually influences the stellar activity and spot alignment. However, the evidence for such star–planet interaction is weak. Shkolnik et al. (2005, 2008) did find hints of planet related chromospheric activity in HD 179949 and υ And. Scharf (2010) found a correlation between X-ray luminosity and the presence of massive, close-in planets. However, Poppenhaeger et al. (2011) could not find any variations in Ca ii and X-rays that could be traced back to the planet in υ And. Furthermore, Poppenhaeger et al. (2010) find that the correlation between X-rays and planet hosting stars can be attributed to selection effects and that in fact no correlation is seen. Thus, there is no compelling observational evidence that the presence of close-in planetary companions influences the activity on the star. Furthermore, these results were for giant planets much more massive than CoRoT-7b. It is not clear how a very low mass planet can influence the activity of the star.
• 3.  
  To produce the symmetrical RV curve in Figure 1 and with little scatter, all four spot groups must have comparable filling factors, otherwise the envelope of RV curves from the individual spot groups would introduce significant "jitter" and the observed scatter of the RV data would exceed the measurement errors. For example, this is seen in the RV curve for Kepler-10 (Batalha et al. 2011) where the assumed RV jitter dominates the internal measurement error in spite of this star being "quiet" in terms of activity. This filling factor can be estimated using the expression of Hatzes (2002) or Saar & Donahue (1997). To produce a spot-induced RV amplitude of 5 m s⁻¹ requires a spot filling factor for each group of about 0.3%. The difference in spot filling factor (areal coverage) can be estimated from the rms scatter about the orbital solution. For the RV curves from the different spot groups to produce variations less than the rms scatter of σ = 1.68 m s⁻¹, they must have the same filling factors to within 70%. Using binned RV values (approximately 5–10 points per bin) in order to minimize the measurement error (≈2 m s⁻¹), one gets an rms scatter of 0.5 m s⁻¹. In this case, the spot-filling factors for all spot groups must be the same to within about 10%.
• 4.  
  A symmetrical RV curve also implies that the spot evolution over the span of the measurements for all four spot groups must be small. The RV measurements span more than 3.5 rotation periods and as noted by Pont et al. (2011): "no feature is reproduced unchanged after one rotation period, and the light curve becomes unrecognizable after merely 2–3 periods." Having four spot groups that evolve very little over 3.5 rotation periods seemingly contradicts what we see in the light curve of CoRoT-7b.

So, either we have a star with a super-Earth planet (similar to Kepler-10b—see below), or we have a star with a remarkable spot distribution. We thus conclude that the 0.85 day RV modulation seen in Figure 1 is due almost entirely to the reflex motion induced by a planet. Taking all the arguments into account, the most reasonable, and most logical, explanation for the 0.85 day RV variations of CoRoT-7 is that these are entirely due to the transiting planet, CoRoT-7b. Furthermore, the significance of this detection is at the 15σ level.

The planet mass can be derived from the K-amplitude via the mass function, f(m) which for circular orbits (the most likely case for CoRoT-7b) is

$$\begin{equation} f(m) = {{ (m_p \sin i)^3 } \over {M_{{\rm star}}^2}} = {{K^3P} \over {2\pi G}}, \end{equation} \tag{ 2 }$$

where P is the orbital period, M_star is the stellar mass, and G is the gravitational constant. Bruntt et al. (2010) derived a stellar mass of 0.91 ± 0.03 M_☉ and Léger et al. (2009) derived an orbital inclination of i = 801 ± 03. Using the appropriate stellar and orbital parameters for CoRoT-7b and solving Equation (1) for m_p results in m_p (M_⊕) = (1.416 ± 0.031)K (m s⁻¹). Our best-fit K-amplitude of 5.25 ± 0.84 m s⁻¹ (linear model) results in m = 7.42 ± 1.21 M_⊕.

Table 3 lists all the different K-amplitude determinations for CoRoT-7b from here and the literature. For Queloz et al. (2009), we include not only the adopted value but also the results from the individual methods for determining the RV amplitude. To ensure that the mass was determined in a consistent way using the same stellar and orbital parameters, we recalculated these using the published K-amplitudes and the mass function. These may result in slightly different values than quoted in the respective papers. These appear in the third column in the table and graphically in Figure 10. The number on the abscissa corresponds to that in the first column of the table. Our mass for CoRoT-7b agrees to within 1σ to most other mass determinations. It is most discrepant (4σ) with the harmonic filtering without correction from Queloz et al. (2009), but the authors showed that the filtering process resulted in an unusually low value. The authors considered the adopted value of 4.96 ± 0.86 to be a more accurate determination.

Zoom In Zoom Out Reset image size

Our mass value is clearly discrepant with the low value of 2.3 ± 1.8 M_⊕ (∼1σ) claimed by Pont et al. (2011), a mass determination largely responsible for casting doubt on the rocky nature of CoRoT-7b (Batalha et al. 2011). The results of Pont et al. were based on a rotating, time-evolving spot model consisting of 2–200 spots with their own latitude, longitude, and scale factor (i.e., temperature). Solar-like differential rotation was also included. The evolution was characterized via a Gaussian with parameters of peak intensity, epoch, and lifetime. Although this model may be a description of activity of a solar-like star in general it is unclear whether this is a realistic model for this star in particular. A large number of free parameters and assumptions are used to fit integrated quantities. One could imagine that with such a complicated model one can use activity alone to fit the RV variations, including those contributed by the reflex motion due to CoRoT-7b. Furthermore, the Pont et al. result represented an average over all possible spot models considered, but we are concerned with the K-amplitude in the presence of the actual spot distribution on CoRoT-7. Our approach, on the other hand, has far fewer assumptions and is largely independent of the exact details of the spot distribution and evolution on the star. We consider that our K-amplitude determination results in more realistic mass for CoRoT-7b.

Given our derived planet mass, we can now estimate the bulk density of the planet. Bruntt et al. (2010) give a planet radius of 1.58 ± 0.1 R_⊕. This results in a mean planet density of 10.4 ± 1.8 gm cm⁻³.

The large density of CoRoT-7b has been reinforced by the discovery of a second transiting super-Earth, Kepler-10b (Batalha et al. 2011). Kepler-10b has a slightly smaller mass, m = 4.56 ± 1.23 M_⊕ and a smaller radius, R = 1.416 ± 0.0345. (For simplicity, we have taken the mean of the ±error values given in Batalha et al. 2011). However, to within the errors CoRoT-7b and Kepler-10b have identical bulk densities: ρ_{Kepler − 10b} = 8.8 ± 2.5 gm cm⁻³ compared to ρ_{CoRoT − 7b} = 10.4 ± 1.8 gm cm⁻³.

What is so striking about these two planets is not only their comparable densities, but that they also orbit very similar host stars (G9V and G4V, respectively) that have a similar radius and mass. The only differences are in the stellar metallicity, Kepler-10 is slightly metal poor with Fe/H] = −0.15 ± 0.04 (Batalha et al. 2011) compared to the CoRoT-7 value of [Fe/H] = +0.12 ± 0.06 (Bruntt et al. 2010), and in the activity level. CoRoT-7 shows relatively high levels of activity, while Kepler-10b is one of the quietest stars in the Kepler sample. This is explained by Kepler-10 also being much older than CoRoT-7, about 8 ± 0.26 Gyr (as derived by asteroseismology). On the other hand, CoRoT-7 has an estimated age of 1.2–2.3 Gyr (Léger et al. 2009). Another similarity between the two systems is that of the very remarkable orbital period of 0.85 day for CoRoT-7b, which was unprecedented for such a low-mass planet, is now mirrored in Kepler-10b with a period of 0.84 day.

Kepler-9d (Holman et al. 2010; Torres et al. 2011) has a similar well-determined radius (≈1.6 R_⊕), a slightly longer period (1.59 days) and an upper mass limit of 7 M_⊕, which, if confirmed, would bring it into the same density regime as CoRoT-7b and Kepler-10b. This is not the case for the small planets recently discovered around the star Kepler-11 (Lissauer et al. 2011), which if their masses are confirmed belong to a type with very much lower densities (≈0.8–2 gm cm⁻³), so there appears to be a wide range of properties of super-Earth exoplanets. This is also reflected in 55 Cnc-e a recently discovered transiting super-Earth (Winn et al. 2011; Demory et al. 2011). Compared to CoRoT-7b and Kepler-10b, this exoplanet has a larger radius of 2.13 ± 0.14 R_⊕, a comparable mass of 7.98 ± 0.69 M_⊕, but half the density at 4.56 ± 0.99 gm cm⁻³. This implies the presence of a significant amount of volatiles.

A detailed discussion on the possible internal structure of CoRoT-7b and Kepler-10b is well beyond the scope of this paper. We can, however, compare the density and radius to terrestrial bodies in our own solar system as well as to simple models. The inner terrestrial planets, excluding Mercury, and the Moon show a tight linear correlation between logarithm of the density, ρ, and planet radius, R_P (correlation coefficient, r = 0.99976). The terrestrial planets as well as the Moon are shown in Figure 11 along with CoRoT-7b and Kepler-10b. Also shown are three models for a "super-Moon," Earth-like, and a "super-Mercury" (a super-Earth with a structure like Mercury). The Earth-like planet has a silicate mantle that is 67% by mass and an iron core that is 10% by mass. A Moon-like planet has a 100% silicate mantle (MgO and SiO₂) and no core. The Mercury-like planet has an iron core that is 63% of the mass and a silicate mantle that is 37% of the planet mass.

Zoom In Zoom Out Reset image size

The nominal values of CoRoT-7b and Kepler-10b place these above the Earth-like planets and close to Mercury that is iron enriched. However, within the errors both of these exoplanets have a density and mass consistent with either an Earth-like planet or a super-Mercury. Whether CoRoT-7b and Kepler-10b have an internal structure similar to Mercury or the Earth requires reducing the error on the density. For CoRoT-7b, equal contributions to the error come from the planet mass (19%) and radius (6%, but contributing as R³ in the density). Kepler-10b benefits from a better measurement of the R_p (due to the asteroseismic determination of the stellar parameters), but the planet mass is known only to within 26%, a bit worse than for CoRoT-7b. A substantial improvement in the mass of Kepler-10b can be realized by a reduction in the K-amplitude error. The RV measurements for Kepler-10b show an rms scatter about the orbit that is a factor of two larger than the internal errors. This is most likely due to the RV jitter of activity (Batalha et al. 2011). This activity jitter is estimated to be about 2 m s⁻¹, or 50% of the planet K-amplitude. So, in order to get a better mass determination for Kepler-10b one would have to correct for the activity signal, even for such a quiet star. Since Kepler-10b has a long rotational period (P_rot = 50–100 days), the stellar activity jitter could be reduced by taking several measurements of Kepler-10 throughout the night when the star is well placed in the sky and employing the procedure we used on CoRoT-7b. Unfortunately, the Kepler-10b RV measurements had only two nights where multiple measurements were taken.

In light of the discovery of Kepler-10b, the properties of CoRoT-7b may not be so extraordinary. It is encouraging that so quickly after the start of the space missions CoRoT and Kepler two transiting rocky planets with comparable (and possibly new) properties have been discovered. This bodes well for the detection of more of such objects.

Figure 11 hints that in terms of structure, CoRoT-7b and Kepler-10b may be more like Mercury than the other terrestrial planets. Clearly, more transiting super-Earths must be found—and with excellent mass and radius determinations—before we know if CoRoT-7b and Kepler-10b just represent a part of the "continuum" of low-mass planets, or whether they are special even among short-period super-Earths.

Not surprisingly, the first confirmed super-Earths found by both CoRoT and Kepler are among the brightest stars in their respective samples. Because they are relatively bright it was possible to get the requisite precise RVs needed to confirm the nature of the transiting object. There are undoubtedly more CoRoT-7b-like planets to be found in the CoRoT and Kepler samples. Unfortunately, these are most likely among the fainter stars of the sample for which the RV determination of the mass is challenging. Clearly, to make significant progress in the understanding of the CoRoT-7b-type planets it is essential to find such objects around relatively bright stars for which characterization studies are easier. Unfortunately, the requisite precise photometry can only be conducted from space. This only emphasizes the need for such proposed space missions as PLAnetary Transits and Oscillations (PLATO; see Catala 2009) and the Transiting Exoplanet Survey Satellite (TESS; see Ricker et al. 2009) whose goals are to find transiting terrestrial planets among the brightest stars. In the case of PLATO, stellar parameters will be determined precisely using asteroseismology which translates into more accurate planetary parameters. We will thus be able to determine if the "CoRoT-7b-like" planets are indeed more like Mercury than the Earth in terms of their internal structure.