On the alignment of debris disks and their host stars’ rotation axis – implications for spin-orbit misalignment in exoplanetary systems
July 26, 2019
It has been widely thought that measuring the misalignment angle between the orbital plane of a transiting exoplanet and the spin of its host star was a good discriminator between different migration processes for hot-Jupiters. Specifically, well-aligned hot-Jupiter systems (as measured by the Rossiter-McLaughlin effect) were thought to have formed via migration through interaction with a viscous disk, while misaligned systems were thought to have undergone a more violent dynamical history. These conclusions were based on the assumption that the planet-forming disk was well-aligned with the host star. Recent work by a number of authors has challenged this assumption by proposing mechanisms that act to drive the star-disk interaction out of alignment during the pre-main sequence phase. We have estimated the stellar rotation axis of a sample of stars which host spatially resolved debris disks. Comparison of our derived stellar rotation axis inclination angles with the geometrically measured debris-disk inclinations shows no evidence for a misalignment between the two.
keywords:planetary systems – stars: activity – stars: rotation
The discovery of planets beyond the confines of our Solar system has presented many surprises and continues to challenge our understanding of planet formation and their subsequent evolution. This is particularly true in the case of hot-Jupiters, whose short orbital periods of a few days or less was unexpected – under the standard core-accretion theory of planet formation, volatile gas-giants should form beyond the snow-line (Pollack et al. 1996). It is now widely accepted that hot-Jupiters did not form in-situ at their current locations, but that some mechanism caused their inwards migration towards their parent star.
A number of theories have been postulated to explain planetary migration. One possible mechanism for forming short-period gas-giants is the pumping of initially wide circular orbits to high eccentricities. This could occur via planet-planet scattering (Rasio & Ford 1996; Weidenschilling & Marzari 1996), or perturbations from a distant stellar binary companion (Eggenberger et al. 2004). The highly eccentric orbit then brings the gas-giant sufficiently close to the host star that tidal dissipation quickly draws the planet to a new, smaller orbital separation. In this scenario, the interactions and scattering involved may lead to large changes in the value of the orbital inclination. Interactions between the planet and a viscous disk, on the other-hand, may also drive the planet inwards but is not thought to perturb the initial orbital inclination.
The close alignment of the rotation and orbital axes in the Solar system (7; Beck & Giles 2005) is attributed to the formation of the Sun and planets from a single rotating proto-stellar disk which was also initially coplanar to the solar-rotation axis. On the premise that disks and stellar rotation axes are aligned, Rossiter-McLaughlin (RM) observations of transiting systems (e.g. Triaud et al. 2010 and references therein) have sought to discriminate between migration caused by planet-disk interactions (leading presumably to aligned systems), and migrations involving some violent dynamical history (leading to misaligned systems). Recent theoretical work, however, challenges the view that the stellar spin axis and the disk rotation axis should be aligned. For example, in numerical simulations of star formation, *bate10 found that the rotation axis of the final disk may be heavily governed by the angular momentum component of the material that was last accreted. While this can lead to a misalignment between the disk and star, it is only thought to be significant for light disks which may have insufficient mass to form giant planets.
On the other-hand, Lai et al. (2010) present arguments that the observed star-orbit misalignment could instead result from alterations in the stellar spin axis, introduced by the star-disk interaction during the pre-main-sequence phase (also see Foucart & Lai 2010). Lai et al. consider the well known fact that a magnetic protostar exerts a warping force on the inner part of the accretion disk (e.g. Bouvier et al., 2007). Previous authors have assumed that this results in significant warps to the inner disk, whereas Lai et al. (2010) argue that viscous processes in the disk itself will smooth these torques, resulting in a largely unwarped inner disk. Given a flat disk, the torques arising from the star-disk interaction will act on the star itself, changing the stellar spin axis on a timescale given by
where and are the mass and radius (in solar units) of the protostar, respectively, is the accretion rate in solar masses per year, is the inner radius of the accretion disk in stellar radii, is the spin rate of the protostar and is the rotation rate of the accretion disk at the inner disk radius.
However, the mechanism proposed by Lai et al. (2010) may not be effective in practice, as the timescale for spin evolution, , is of the same order as the disk evolution timescale. Near-infrared observations of protostars show that the majority of protostellar disks have dispersed by the age of 5 Myr (Hernández et al., 2008) whilst observations of the accretion rates onto young stars also show that the accretion rate declines rapidly with increasing age and decreasing stellar mass (e.g Sicilia-Aguilar et al., 2006), so the accretion rate on many protostars may well be below the canonical Myr assumed by Lai et al. (2010). In addition, the results of Lai et al. (2010) rely on the inner disk not being ‘significantly warped’, however there is good evidence that the inner disks of some young stars do contain significant disk warps (see e.g. Bouvier et al., 2007; Muzerolle et al., 2009).
If correct, these theoretical papers potentially have important ramifications for our interpretation of the results of RM observations. Indeed, if the stellar rotation axis can be driven from coplanarity with the surrounding disk, or vice-versa, then RM observations would essentially be rendered useless as a tool for determining the migration mechanism responsible for forming hot-Jupiter’s. For these reasons, it is important to seek observational evidence for such processes. In this paper we present a study of star-disk alignment in debris disk systems.
2 Measuring the star-disk alignment
For the purposes of this work, we have concentrated on systems with spatially resolved debris disks. We have also assumed that the debris-disk plane is representative of the primordial disk and, likewise, that the presently observed stellar orientation is the same as the protostars’. The inclination of the disk to our line-of-sight can then be measured geometrically by calculating the fore-shortening of the semi-minor axis of the disk relative to the semi-major axis (although in reality the models used to determine the disk geometry are somewhat more complex).
A more indirect approach is needed in order to determine the inclination angle of the stellar rotation axis, however. To do this we have followed the method of Watson et al. (2010) who compiled the stellar rotation inclination angles for 117 exoplanet host stars, and we refer the reader to that paper for in-depth details of the methods used, as well as a discussion on possible sources of systematic errors inherent in the technique. In summary, it is possible to determine the inclination angle, , between the rotation axis of a star and the observers line-of-sight from measurements of the projected equatorial velocity (), the stellar rotation period () and the stellar radius () via the equation
The projected equatorial rotation velocity, , can be measured using high-resolution spectroscopy, while the stellar radius can also be indirectly determined from spectra or, less frequently, directly via interferometry, lunar occultations or eclipses (e.g. Fracassini et al. 2001). Precisions on stellar radius measurements of 3 per cent are now regularly quoted (e.g. Fischer & Valenti 2005).
Determining the stellar rotation period, on the other-hand, tends to be more troublesome. For some active stars, the stellar spin period can be determined photometrically to high precision by tracking the passage of large star spots on their surfaces. For those systems which do not have photometrically measured rotation periods, measurements of Ca ii H and K emission can be used to estimate the rotation period by applying the chromospheric emission – rotation period relationship of Noyes et al. (1984). Naturally, this latter method is less precise, and is also affected by intrinsic variability of the Ca ii H and K emission due to, for example, solar-like activity cycles or the rotation of magnetic regions.
We have carried out an extensive literature search and present , , and estimates for a number of main-sequence stars which host spatially resolved debris disks in Table 1. Since one of the pre-requisites for measuring a stellar rotation period is that the star must be magnetically active, we are restricted to lower main-sequence stars later than F5V which have a convective envelope (and are thereby capable of sustaining a stellar dynamo). Of the 20 main-sequence stars with resolved debris disks, only 10 have spectral types of F5V or later. Of these, we can find no recorded Ca ii H and K emission measurement for HD 181327, and is therefore omitted from our list.
We should note that we have not considered pre-main sequence stars in our analysis. This is for two principal reasons. First, given their fully convective nature, it is not certain that the activity-rotation period relationship of Noyes et al. (1984) (which was calibrated for main-sequence stars) holds, indeed an entirely different stellar dynamo mechanism may operate in pre-main sequence stars (e.g. Scholz et al. 2007). Second, radius estimates for pre-main sequence stars are also notoriously unreliable, since they depend upon age estimates which are uncertain by a factor of several (e.g. Naylor 2009; Baraffe, Chabrier & Gallardo 2009).
2.1 Adopted stellar parameters and errors
In order to determine via equation 2, we have taken a weighted mean of the entries in Table 1 for the final values of and . In identical fashion to that carried out in Watson et al. (2010), where no error was quoted on a published value we have taken it to be 1.0 kms (twice the typical error assumed on measurements, see the catalogue of Fischer & Valenti 2005 for example). Regarding published radii with no associated error bar, we have taken the error to be 10 or 20 per cent of the absolute value. The choice between 10 or 20 per cent is taken to ensure that radii estimates with associated error bars were given a higher weighting than those without formal errors.
For stars with photometrically derived rotation periods which have no associated error bar, we have taken the error to be 10 per cent. This is commensurate with the typical error bars quoted on such measurements. Where available, photometrically derived rotation periods are adopted, otherwise the rotation period is estimated from the strength of the Ca ii H and K emission (Noyes et al. 1984). Again, following Watson et al. (2010), for each measurement reported in Table 1 we have determined, where possible, the number of observations and period over which they were carried out (see Table 4). Where details are not present, or are ambiguous, we have assumed they are from a single observation and have flagged them as ‘individual?’. As in Watson et al. (2010), each star was assigned a grade of P (Poor), O (O.K.), G (Good) or E (Excellent) based on how well monitored it was. We then assigned general error bars on the values dependent on their assigned grades and spectral type. These error bars are derived from the average rotationally modulated variations outlined in Section 3.1 of Watson et al. (2010). For a detailed discussion of the systematic errors on the derived parameters, we refer the reader to this work. Table 2 lists the adopted parameters for each star.
2.2 Determining the stellar inclination angle
Equation 2 can be thought of as a naive estimator of as it is geometrically unconstrained (e.g. 1 is allowed). While a value of 1 is unphysical, it does allow potential problem cases to be identified. Again, we follow Watson et al. (2010) and reject systems with ’s that are 1- greater than 1 from further analysis – flagging these as having a high probability of being affected by systematic errors. This results in the omission of 2 systems, HD 53143 and HD 139664, both of which have naive estimates significantly greater than 1 (see the first two entries of Table 2). In the case of HD 139664, the value places it at the extreme edge of the chromospheric emission – rotation period calibration by Noyes et al. (1984). In addition, the star is classified as having a luminosity class IV, and therefore both the derived rotation period from the Noyes et al. (1984) relationship (which is only calibrated for main-sequence stars) and radius may also be suspect. HD 53143, on the other hand, is more problematic. It appears to have a secure rotation period which has been measured photometrically and that also agrees very well with the period derived from the Ca ii H and K emission. In addition, all of the measured ’s and radii are consistent with one another. Yet, despite this and the fact that it appears to be a solid main-sequence star with an age of 1.0 0.2 Gyr (Kalas et al. 2006), we derive 1.5 0.4. We can only assume that 1 or more of the measurements are affected by systematics.
For the 8 remaining systems we have carried out a Markov-chain Monte Carlo (MCMC) analysis which not only provides a means of optimising the fit of a model to data but explores the joint posterior probability distribution of the fitted parameters and allows proper 1- two-tailed confidence limits to be placed on the derived ’s. In addition, MCMC rejects unphysical combinations of parameters that result in 1. For the purposes of this work, we have followed the MCMC process outlined in Watson et al. (2010), keeping the same 1000-step burn-in phase and carrying out 1,000,000 jumps. The results of this MCMC analysis are shown in Table 2.
3 Results and discussion
Table LABEL:tab:comparison shows our derived stellar rotation inclination angles versus published debris-disk inclinations. As can be seen, there is no obvious evidence for large mis-alignments of the stellar rotation axes and debris-disk planes in any of these systems. By the nature of the method, the best constrained systems have 0.5 ( =30). This is because at high inclinations the sine curve is relatively flat, and thus small errors in (which is what is directly calculated from the observables in equation 2) propagate to form large errors when expressed in degrees. At ’s of 0.5, the sine curve is much steeper, and travelling along the sine curve does not vary the inclination as quickly as it does at high ’s. As one moves to lower ’s, measurement errors on naturally increase as the projected rotational broadening decreases. The fact that the best constrained systems, HD 22049 and HD 107146, with errors on of only 5 – 9 appear to align closely with their debris disk gives us both confidence in the technique, and further strengthens our assertion that we see no evidence for a detectable difference between the sky-projected angle of the disk and the that of the stellar rotation axis. In addition, it should be noted that HD 22049 is known to host a planet that has had the inclination of its orbital plane accurately determined to be = 30.13.8 – suggesting coplanarity between the planetary orbit and disk (Benedict et al. 2006). Furthermore, star spot modeling of a MOST light curve of HD 22049 by Bryce et al. (2006) determined the inclination of the stellar rotation axis to be = 303, in excellent agreement with our derived values. We do caution, however, that the absolute direction of the axis (whether the rotation axis is pointing towards or away from the observer) cannot be ascertained, and therefore we do not have a knowledge of the full three-dimensional geometry of the star-disk systems.
We can test the significance of our result using a rank-order approach. If the disk and stellar spin axes are closely aligned, we expect a ranking by disk inclination to agree well with a ranking by stellar inclination. This is indeed what we find. We calculate the spearman rank-order correlation coefficient for our data, and find a value of 0.82. If we repeat this analysis for all 40320 random permutations of our stellar spin data, only in 0.1% of cases do we find a better correlation between stellar spin and disk inclinations. This suggests we reject the null hypothesis (that stellar spin and disk inclinations are uncorrelated) at a significance of 99.9%.
We performed a further test of significance by first assuming that the stellar spin and disk axes were uniformly distributed around the sky. We drew 8 values of disk and stellar inclinations from a uniform distribution in , and took account of our errors by perturbing these inclinations by an amount equal to the errors on our observations. For the inclinations, which have asymmetric errors, we used an average of the two errors. For the disks we assigned an error of 3 degrees where none was quoted. Where the given disk inclination is a range, we set the error to half of that range. We computed such simulated datasets, and only 0.4% showed a better correlation (as judged by the Spearman rank-order coefficient) than our original data. We therefore reject the hypothesis that the stellar spin axes are independent of the disk inclination axes with a significance of 99.6%.
In reality, it is likely that the stellar spin axes and disk axes are aligned to some degree of precision. It is reasonable to ask what is the largest degree of misalignment permitted by our data. Answering such a question would require a full Bayesian treatment which accounts for the fact that we only observe the component of misalignment along the line of sight. Such an analysis is quite complex (see Fabrycky & Winn, 2009, for example), and is left for a future work.
A recent analysis of Rossiter-McLaughlin observations by Triaud et al. (2010) suggest that between 45 – 85 per cent of hot-Jupiters appear to be significantly misaligned. Our work in this paper reveals no similar degree of misalignment between debris disks and their host stars. We note, however, that all of the systems in our study have host stars with effective temperatures below 6140K (see Table LABEL:tab:comparison). Recently, Winn et al. (2010) have highlighted that exoplanet host stars with effective temperatures below 6250K appear to have the planet-star spin axes preferentially aligned, whereas exoplanets orbiting hotter host stars are more likely to be misaligned. It is, therefore, possible that a yet to be determined mechanism which only drives star-disk misalignments in hotter systems could be operating which our small study has missed. We conclude, however, that there appears to be no substantial evidence to suggest that a universal process, such as that outlined by Lai et al. (2010) and Bate et al. (2010), is a major mechanism in misaligning planetary orbits.
References: NStED, Valenti & Fischer (2005), Wright et al. (2004), Nordström et al. (2004), Henry et al. (1996), Reiners & Schmitt (2003), Glebocki & Stawikowski (2000), Coralie, Jenkins et al. (2006), Torres et al. (2006), *schroder09, Rhee et al. (2007), Allende Prieto & Lambert (1999), Takeda et al. (2007), White et al. (2007), Gray et al. (2006), Fracassini et al. (2001), Pizzolato et al. (2003), Ochsenbein & Halbwachs (1999), Pojmański & Maciejewski (2005), Samus et al. (2009), Baliunas et al. (1996), Noyes et al. (1984), Mallik et al. (2003), Saar & Osten (1997), Simpson et al. (2010), Fray et al. (1991)
: = value uncertain. = mean of a range of values given by Glebocki & Stawikowski (2000). = rotation period measured photometrically.
|10647||49||52||(Liseau et al. 2008)||1.20||6140|
|10700||45||60–90||(Greaves et al. 2004)||0.91||5500|
|22049||31||25||(Greaves et al. 1998)||0.78||5090|
|61005||90||80||(Maness et al. 2009)||0.89||5440|
|92945||65||70||(Krist et al. 2005)||0.77||5060|
|107146||21||255||(Ardila et al. 2004)||1.09||5850|
|197481||90||90||(Krist et al. 2005)||0.49||3560|
|207129||47||60||(Krist et al. 2010)||1.11||5890|
|-4.700||individual on 2001 Aug 04||9.|
|(P) Adopted value: -4.698 0.060|
|HD 61005||G3/5V||-4.260||1 obs on UT 14/12/1992||5|
|-4.360||1 obs on 28/10/2002||15|
|(P) Adopted value: -4.320 0.075|
|HD 92945||K1V||-4.320||13 obs in 6 months. Report = 2.72%||3|
|(O) Adopted value: -4.325 0.077|
|HD 107146||G5||-4.340||8 obs in 5 months. Report = 3.04%||3|
|(O) Adopted value: -4.340 0.057|
|(P) Adopted value: -4.621 0.060|
|HD 207129||G0V||-4.800||1 obs on UT 28/06/1993||5|
|-4.850||1 obs on 2004 Aug 23/24||9|
|(P) Adopted value: -4.89 0.075|
CAW and CD would like to thank the Nuffield Foundation for funding a science bursary for CD to undertake a summer placement at QUB. SPL acknowledges the support of an RCUK fellowship. Finally, we would like to thank Dan Fabrycky for his comments which helped to substantially improve the conclusions of this paper.
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