The coronal X-ray - age relation and its implications for the evaporation of exoplanets
aa r X i v : . [ a s t r o - ph . E P ] A p r Mon. Not. R. Astron. Soc. , 1–20 (2011) Printed 24 October 2018 (MN L A TEX style file v2.2)
The coronal X-ray – age relation and its implications for theevaporation of exoplanets
Alan P. Jackson , ⋆ , Timothy A. Davis , ⋆ and Peter J. Wheatley Sub-Department of Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, UK, OX1 3RH Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, UK, CB3 0HA European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748, Garching bei Muenchen, Germany Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, UK, CV4 7AL
Submitted 2011
ABSTRACT
We study the relationship between coronal X-ray emission and stellar age for late-type stars,and the variation of this relationship with spectral type. We select 717 stars from 13 openclusters and find that the ratio of X-ray to bolometric luminosity during the saturated phaseof coronal emission decreases from − . for late K-dwarfs to − . for early F-typestars (across the range . ( B − V ) < . ). Our determined saturation timescales varybetween . and . years, though with no clear trend across the whole FGK range.We apply our X-ray emission – age relations to the investigation of the evaporation historyof 121 known transiting exoplanets using a simple energy-limited model of evaporation andtaking into consideration Roche lobe effects and different heating/evaporation efficiencies. Weconfirm that a linear cut-off of the planet distribution in the M /R versus a − plane is anexpected result of population modification by evaporation and show that the known transitingexoplanets display such a cut-off. We find that for an evaporation efficiency of 25 per cent weexpect around one in ten of the known transiting exoplanets to have lost > per cent of theirmass since formation. In addition we provide estimates of the minimum formation mass forwhich a planet could be expected to survive for 4 Gyrs for a range of stellar and planetaryparameters.We emphasise the importance of the earliest periods of a planet’s life for it’s evaporationhistory with 75 per cent expected to occur within the first Gyr. This raises the possibilityof using evaporation histories to distinguish different migration mechanisms. For planets withspin-orbit angles available from measurements of the Rossiter-McLaughlin effect no differenceis found between the distributions of planets with misaligned orbits and those with alignedorbits. This suggests that dynamical effects accounting for misalignment occur early in the lifeof the planetary system, although additional data is required to test this. Key words: stars: late-type – stars: activity – X-rays: stars – planetary systems – planetarysystems: formation
Since the discovery of the first exoplanet (51 Peg. b) around amain sequence star by Mayor & Queloz (1995), a roughly Jupitermass planet in a ∼ ⋆ E-mail: [email protected] (APJ); [email protected] (TAD)
Mazeh, Zucker & Pont (2005) and Southworth, Wheatley & Sams(2007) respectively. Both correlations have subsequently beenre-plotted by a number of authors as more exoplanets have beendiscovered (e.g. Hansen & Barman 2007; Pollacco et al. 2008;Davis & Wheatley 2009) and are reproduced in Section 3.1 for thesample of exoplanets used in this work.One possible mechanism for producing such correlations isthe evaporation of close orbiting planets, such as that observedfor HD209458b (also known as Osiris, e.g. Vidal-Madjar et al.2008) by Vidal-Madjar et al. (2003) and HD189733b byLecavelier des Etangs et al. (2010).A simple model of exoplanet evaporation was developed by see also Ben-Jaffel (2007); Vidal-Madjar et al. (2008)c (cid:13) A.P. Jackson, T.A. Davis and P.J. Wheatley
Lecavelier des Etangs (2007), driven by X-ray/Extreme Ultraviolet(EUV) radiation. He applied it to the known exoplanets and foundthat they should not be losing significant mass at today’s irradiationlevels. The X-ray/EUV irradiation level will be much higher arounda younger solar-type star however, as discussed further below,and Penz, Micela & Lammer (2008) showed that, in principle, it ispossible for evaporation to significantly effect the mass distributionof close orbiting exoplanets.As already noted by Lecavelier des Etangs (2007) andPenz et al. (2008) it is clear that to understand the evaporationof close orbiting exoplanets an understanding of the X-ray/EUVemission of the host star and its evolution is essential.Davis & Wheatley (2009) and others such as Lammer et al.(2009) (hereafter DW09 and L09 respectively) combined theenergy-limited model developed by Lecavelier des Etangs (2007)with simple, coarse, estimates of the X-ray evolution to study thepossible evaporation histories of known transiting exoplanets.The majority of the X-ray/EUV luminosity of a star isemitted from the hot corona (with typical temperatures of order1 keV), the heating of which is believed to be related tomagnetic activity within the star (e.g. Erd´elyi & Ballai 2007).The stellar magnetic fields, of which this activity and X-rayemission is a manifestation, are themselves thought to be derivedfrom complex dynamo mechanisms at work in the interior ofthe star. The efficiency of these mechanisms is determined bythe interaction between convection in the outer envelope anddifferential rotation (e.g. Landstreet 1992; Covas, Moss & Tavakol2005; Donati & Landstreet 2009). Skumanich (1972) observeda proportionality between average surface magnetic field andstellar rotation rate and was the first to suggest a relationshipbetween activity and rotation as a consequence of the dynamomechanism. Since then many authors, such as Pallavicini et al.(1981), Randich et al. (1996) and Pizzolato et al. (2003) (hereafterP03), have studied the relationship between activity/X-ray emissionand rotation.Over time the rotation rate of a star slows as a resultof magnetic braking (e.g. Ivanova & Taam 2003) and since themagnetic dynamo is linked to the stellar rotation the magnetic fieldwill also decrease over time. Thus it would be expected that theX-ray luminosity will fall with decreasing rotation rate and indeedthis is observed, e.g. P03 see a decrease for periods longer than ∼ ∼ To obtain constraints on the evolution of X-ray emission fromsolar type stars we require stars of known age across a range ofages. Therefore we have selected stars from open clusters withage estimates, the clusters being selected such that they cover arange of ages from ∼ − Myr. We utilise open clusters sincethey form in a short space of time, and thus all of the memberswill be of similar age. Details of the clusters can be found inTable 1 and a complete catalogue of the stars used in this work isavailable online through the VizieR database . The star/cluster datais taken from past X-ray surveys of open clusters in the literature.Most of the original observations were taken using the PSPC orHRI instruments on ROSAT (R¨ontgen Satellit) , though some ofthe more recent surveys were conducted using the
Chandra and
XMM-Newton telescopes.Within the datasets for each cluster we select stars that fall inthe range . ( B − V ) < . (roughly stars of spectral typeF, G and K), which have well defined X-ray luminosities (i.e. notupper limits). We exclude stars with only upper limits since theseare presented inhomogeneously or not at all in the sources fromwhich we draw our sample, and a meaningful comparison betweenthese estimates could not be made. Where cluster membershipinformation is given we require the stars to be likely clustermembers/have a membership probability of > for selection.In addition to the data from the open clusters listed in Table 1we also include in our plots the field star sample of P03 as a guideto how our results relate to the properties of stars of Solar age. Wedo not use these stars in any of the fitting however. A list of the starsin this field star sample can be found in Table 1 of P03 Where X-ray luminosities have been calculated in the sourcepapers, as they are for all of the clusters except NGC 6530, weuse the luminosities given. For NGC 6530 we use the web
PIMMS tool available at NASA HEASARC to determine the count rateto flux conversion factor. The column density of hydrogen, N H ,is obtained from the cluster E ( B − V ) using the formula N H =[ E ( B − V ) × . × ] cm − (Bohlin et al. 1978). Once we havethe cluster distance the X-ray luminosities can then be determined.With the exception of the Pleiades and Hyades (which haveparallax distances) we calculate the cluster distance from theapparent distance modulus in conjunction with the reddening, E ( B − V ) , and the extinction law A V = 3 . × E ( B − V ) (e.g.Schultz & Wiemer 1975; Sneden et al. 1978). Where these distanceestimates (from apparent distance modulus or parallax) differ fromthose used in the original study, and are based on more recent data,we correct the X-ray luminosities for the new distance.Spectral type information is not available for the majorityof stars used in this work so we use the de-reddened colourindices, ( B − V ) = ( B − V ) − E ( B − V ) , to assignspectral types using the tables presented in Lang (1991). Wherenecessary we calculate bolometric luminosities using the absolutemagnitude of the star. We determine the absolute magnitudefrom the apparent V magnitude and the cluster apparent distancemodulus in combination with a bolometric correction calculated VizieR data web address heasarc.gsfc.nasa.gov c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation Table 1.
Details of the open clusters used in this work and the source of the stellar data for each.Cluster Age (Myr) Distance (pc) Apparent distance E ( B − V ) No. of Stellar data sourcemodulus (mag) (mag) stars α Persei 79 a,b,c a,b a,b a,b
66 Randich et al. (1996), P03Blanco 1 100 a,d a,b a a
29 Cargile et al. (2009)Hyades 630 b,e e b b
82 Stern, Schmitt & Kahabka (1995)IC2391 50 a,b,c a,b a,b a,b
19 Patten & Simon (1996), P03IC2602 37 a,b,f a,b a,b a,b
26 Randich et al. (1995), P03NGC1039 192 a,b a,b a,b a,b a,b a,b a,b a,b
97 Damiani et al. (2003)NGC3532 308 a,b a,b a,b a,b a,b,g,h a,b,g,h a,b a,b
41 Jeffries et al. (2006)NGC6475 197 a,b a,b a,b a,b
59 Prosser et al. (1995)NGC6530 5.4 a,b,i a,b a,b a,b
171 Damiani et al. (2004)Pleiades 132 a,b k -- 0.025 a,b
95 Stauffer et al. (1994)Praesepe 741 a,b a,b a,b a,b
14 Randich & Schmitt (1995)
References : a) Kharchenko et al. (2005), b) Loktin, Gerasimenko & Malysheva (2001), c)Barrado y Navascu´es, Stauffer & Jayawardhana (2004), d) Cargile, James & Platais (2009), James et al. (2009, in prep.), e)Perryman et al. (1998), f) Stauffer et al. (1997), g) Naylor et al. (2002), h) Jeffries & Oliveira (2005), i) Damiani et al. (2004), k)Soderblom et al. (2005).Where more than one reference is given the value shown is a weighted mean of the reference values. Where the distance, apparentdistance modulus and E ( B − V ) references are the same the distance is calculated from the apparent distance modulus and E ( B − V ) . using the de-reddened colour index, ( B − V ) , and the tablespresented in Lang (1991). The luminosities are calibrated using anabsolute bolometric magnitude of +4.75 for the Sun, as given inLang (1991). We thus calculate the X-ray to bolometric luminosityratio where this has not been done in the source paper. Within each ( B − V ) bin for each cluster there will be (providedthat there is more than one star present in that bin) a scatter in theX-ray to bolometric luminosity ratios. There will also be errors inthe luminosity ratios of the individual stars and these may be similarin magnitude to the intrinsic scatter.Where errors in the stellar X-ray luminosities are given inthe source papers we use them as provided, where errors are onlygiven in the count rates or X-ray luminosities are not calculatedin the source papers we estimate the error by propagating throughthe count rate and distance errors. As E ( B − V ) and N H arecorrection factors, and are for our sample generally quite small withfairly small fractional errors, we neglect errors in these quantities inour error analysis. The X-ray luminosity errors are thus determinedsolely by the count rate and distance errors.In the case of bolometric luminosities there will be acontribution from the error in the cluster apparent distance modulusand also contributions from errors in the apparent V magnitudeand B − V colour (as before we neglect errors in E ( B − V ) ).Unfortunately however the only cluster for which we have dataon errors in V magnitude and B − V is Blanco 1. As such, andwith the Blanco 1 errors as a rough guide, we decide the mostreasonable course is to estimate the error in the V magnitude and B − V colours of the other clusters as being ± in the last digit.The estimated B − V colour error gives an estimate of the errorin the bolometric correction. The error in the bolometric luminositywill thus be determined by the error in the cluster apparent distancemodulus and the estimated V magnitude and bolometric correctionerrors.Having calculated the errors in the X-ray and bolometric luminosities we can determine an error in the luminosity ratio foreach star and then calculate the mean stellar error within each ( B − V ) bin for each cluster. We then combine this with thestandard deviation of the stars from the mean luminosity ratio inthe relevant ( B − V ) bin of the cluster to determine a totalluminosity ratio error within each ( B − V ) bin for each cluster.We note that for the most well populated clusters this may leadto a slight over-estimation of the errors since some of the scatterbetween different stars in the cluster will be due to the errors inthe determination of X-ray and bolometric luminosities for theindividual stars. For many of the clusters in our sample howeverthere are not enough stars for the cluster error to fully accountfor the errors in individual stars. To ensure that all of the clustersare treated equally we thus combine the cluster and individual starerrors for all clusters. Any over-estimation of the errors for the mostwell populated clusters should be accounted for by the weightingdescribed in Section 2.2.2. As the X-ray surveys are complete surveys of the entire clustersthey do not make any distinctions regarding main-sequence (MS)stars, pre-main sequence (PMS) stars, post-main sequence giants orvariables. We consider that the processes leading to X-ray emissionin PMS stars are likely to be similar to MS stars and thus do notexclude them, indeed in our younger clusters the majority of starswill be PMS stars. Nonetheless we do take note of different classesof PMS stars, in particular variables, and discuss the potentialeffects of these stars on the mean X-ray characteristics in Section2.2.4.On the other hand we would not expect the dynamo processesthat drive coronal X-ray emission to be the same in giants as in MSstars and thus would not expect them to display the same X-rayproperties. As such giant stars should be excluded from our sample.Giant stars are also comparatively easy to identify even where theclusters have not been well studied at other wavelengths due totheir greater luminosity. We thus exclude any star which would c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley have a bolometric luminosity >
100 times that of the Sun if it isat the associated cluster distance. As well as giants this will alsohelp to filter out foreground contaminants of the cluster sample.For variable stars we want to exclude contact/interacting binarysystems, such as W UMa type systems, since it is likely that theinteraction between the stars will have significant effects on theX-ray characteristics.Given the size of the cluster datasets, and that some of ourclusters have not been well studied as yet, it was not possibleto perform checks on all of the stars in each cluster to indentifystars falling into one of the above exclusion criteria. As suchwe chose to check the Pleiades, Praesepe and Hyades datasets asthese clusters are the most well studied at other wavelengths inour sample and use these to inform subsequent targeted checkson the other datasets. The checking process consisted of takingthe star designation/coordinates from the survey data, querying theSIMBAD database and analysing the details returned. From thischecking process we found that in general stars falling into anexclusion category are outliers, and did not show any preferentialdistribution when mixed in with the main bulk of the clusters. Assuch for the remaining clusters we focused checks on stars whichappear to be outliers. As already mentioned however some of theother clusters in our sample have not yet been well studied and solack the detailed information on member stars necessary to identifyexclusion candidates. None of our exclusion criteria result in theexclusion of a very large number of stars though ( < from thePleiades, Praesepe and Hyades in total for all exclusion criteria) sothe lack of information to identify exclusion candidates in someof the clusters should simply increase the scatter in the affectedclusters and not alter the mean values significantly.In addition to the above general categories we also excludestars that are identified as flaring during the observations, sinceby definition a flare will increase the X-ray luminosity of the starsignificantly above the quiescent level. Significant contamination ofthe sample by flares would thus cause the mean X-ray luminositiesto be over-estimated as well as increasing the scatter in ourrelations. Where available we use studies of flare stars to excludethose stars that were in flare during the observations. For thePleiades we use the study of Gagne, Caillault & Stauffer (1995),for NGC2516 we use Wolk et al. (2004) and for NGC2547 weuse Jeffries et al. (2006). We do not exclude ’flare stars’ as ageneral class however and discuss the effect this may have on themean X-ray characteristics in Section 2.2.4. Gagne et al. (1995) andJeffries et al. (2006) estimate a flaring rate of approximately oneflare every 350 ks for flare stars of ages between that of NGC2547and the Pleiades. We thus expect that contamination by flares willnot be too great for clusters at these ages or older since flaringis more strongly associated with PMS stars, and we expect all ofthe stars in these clusters to have joined the MS. Damiani et al.(2004) identify a higher flare rate in NGC6530, the youngest cluster,and also identify the highest flare rates with the youngest stars,though they do not describe individual flares in detail. As well asutilising these studies we exclude any star found to have an X-rayto bolometric luminosity ratio of > − . as highly likely to be inflare, as this is well above the saturated level found by any author. simbad.u-strasbg.fr/simbad; see also Wenger et al. (2000) l og ( L X / L bo l ) (B-V) (mag.) Pizzolato et al. saturation levels Young clusters (5-80Myrs) Blanco 1 (100Myrs) Pleiades, NGC2516 (120-130Myrs) NGC6475, NGC1039 (190-200Myrs) NGC3532 (300Myrs) Old clusters (630-750 Myrs)
Figure 1. log( L X /L bol ) against ( B − V ) colour for all clusters in thiswork and the saturation levels of P03. Clusters of similar age and behaviourare grouped for ease of interpretation. We expect that the behaviour of the X-ray luminosity will changewith stellar mass and with bolometric luminosity. Thus for ouranalysis of the X-ray evolution we divide our data into bins by ( B − V ) colour as a measurable, continuous, proxy for theseproperties. In Fig. 1 we plot log( L X /L bol ) , where L X and L bol are the X-rayand bolometric luminosities, against ( B − V ) colour for all ofthe clusters used in this work along with the saturation levels ofP03. The 5 young clusters, NGC6530, NGC2547, IC2602, IC2391and Alpha Persei, show very similar behaviour in bins that containenough stars to determine a mean log( L X /L bol ) (i.e. one basedon more than 3 stars). As such we group these clusters together asthe ‘young clusters’ in Fig. 1 for clarity. Given the very young ageof NGC6530 and the relatively large age spread of these clusterswe consider the young cluster line to be a good first estimate ofthe X-ray saturation levels across the ( B − V ) range consideredin this work. In the ( B − V ) range covered by P03 this initialapproximation to the saturation levels is similar to, if slightly lowerthan, their levels and continues the trend for the saturation level todecrease for earlier spectral types. Fig. 1 shows in broad terms ageneral trend in the falling off of X-ray emission as a function oftime in combination with lower relative X-ray emission of earliertype stars within the same age group.We note that the Pleiades and NGC2516 have large errors in log( L X /L bol ) in the central bins. This is the result of a bimodalbehaviour of the stars in these clusters within the ( B − V ) rangeof these bins. We believe this behaviour is a feature of the age ofthese clusters and is discussed further in Section 2.2.5. c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation l og ( L X / L bo l ) log (age(yr)) a) l og ( L X / L bo l ) log (age(yr)) b) l og ( L X / L bo l ) log (age(yr)) c) l og ( L X / L bo l ) log (age(yr)) d) l og ( L X / L bo l ) log (age(yr)) e) l og ( L X / L bo l ) log (age(yr)) f) l og ( L X / L bo l ) log (age(yr)) g) Figure 2.
X-ray to bolometric luminosity ratio against age for the open clusters used in this work. Arrows indicate clusters for which 3 or fewer of the selectedstars fall within the relevant ( B − V ) bin, these clusters receive a lower weighting in the fitting procedure. Solid lines indicate the fits to the data with dashedlines as fits that are less certain. We include the field star sample of P03 (marked in red) at an assumed age of 4.5 Gyr as a guide but do not include them in anyof the fitting. See Sections 2.2.2 and 2.2.3 for further information.c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley
Table 2.
Results of the fits to the saturated and non-saturated regimes for each ( B − V ) colour bin. Note that althoughthe . ( B − V ) < . bin has many more stars than most bins almost half of these are from the same cluster,NGC6530. ( B − V ) colour range No. of stars log( L X /L bol ) sat log τ sat (yrs) Power law index, α ( B − V ) < − . ± . ± .
50 7 . ± . ± .
35 1 . ± . ( B − V ) < − . ± . ± .
39 8 . ± . ± .
28 (8 .
30) 1 . ± . ( B − V ) < − . ± . ± .
34 7 . ± . ± .
25 1 . ± . ( B − V ) < − . ± . ± .
47 8 . ± . ± .
31 1 . ± . ( B − V ) < − . ± . ± .
36 7 . ± . ± .
22 1 . ± . ( B − V ) < − . ± . ± .
37 8 . ± . ± .
29 (8 .
27) 1 . ± . ( B − V ) < − . ± . ± .
35 8 . ± . ± .
31 (8 .
21) 1 . ± . Values in parentheses indicate alternative turn-off ages obtained using a fixed gradient of -1 for the fit to the unsaturatedregime.
In Fig. 2 we plot the ratio of X-ray to bolometric luminosity againstage for all of the clusters in our sample for each ( B − V ) bin alongwith our fits to the saturated and non-saturated regimes. The resultsare summarised in Table 2 and Fig. 3.In each ( B − V ) bin we fit a broken power law to the clusters.This is done using an iterative implementation of the methoddescribed by Fasano & Vio (1988) with a slight modification totheir weighting factor, W i , by the addition of a factor C i to thenumerator. In each bin the factor C i is equal to the number of starsin the i th cluster in that bin, normalised to the total number of starsin the bin. This is to account for the fact that clusters with fewerstars in a given bin tend to have a smaller luminosity ratio spread asa result of under-sampling of the distribution.For the fitting the saturated regime is constrained to behorizontal, while the slope of the unsaturated regime is allowedto vary. In the case of the . ( B − V ) < . , . ( B − V ) < . and . ( B − V ) < . bins howeverthere are relatively few clusters lying in the unsaturated regime.As such for these bins we also used a fit with the slope of theunsaturated regime fixed at -1. An unsaturated slope of -1 is whatone would expect as a result of the inverse square root decay ofrotational frequency with time found by Skumanich (1972) and the L X /L bol ∝ ω relation (where ω is rotational frequency) foundby e.g. P03 and Pallavicini et al. (1981). The turn-off ages obtainedfrom these fixed slope fits are included in Table 2 as the values inbrackets and are very similar to the turn-off ages obtained from thevariable slope fits.For both the saturation level and saturation turn-off age wequote two errors. The first error is the error in the position ofthe saturated level and saturation turn-off age given by the fittingprocedure. The second is the root mean square scatter of theindividual stars about these mean positions as a measure of theintrinsic scatter in the relations. The error in the gradient of theunsaturated regime is taken from the fitting procedure similarly tothe first saturation level and turn-off age errors and we note thatthe stellar scatter about the unsaturated regime line is of similarmagnitude to the scatter about the mean saturated level. ( B − V ) dependence of the saturated and non-saturatedX-ray emission In Fig. 4 we plot both our saturated luminosity ratios and thoseof P03 for comparison. Our saturated luminosity ratios are inreasonable agreement with those found by P03, if in general slightlylower, and fall off consistently as one moves to lower ( B − V ) . g)f)e)d)c)b) l og ( L X / L bo l ) log (age(yr)) a) Figure 3.
Summary of the fits from Fig. 2. Each line is marked with theletter of the Fig. 2 plot from which it is taken. See Sections 2.2.2 and 2.2.3for further information. (L X /L bol ) sat , this work (L X /L bol ) sat , P03 l og ( L X / L bo l ) s a t (B-V) (mag.) Figure 4.
Saturated X-ray to bolometric luminosity ratio for our workand that of P03 against ( B − V ) colour. The capped error bars for oursaturated luminosity ratio correspond to the error in the placement of themean saturated level with the dotted capless error bars corresponding to thetypical spreads about the saturated level. c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation (B-V) (mag.) l og10 ( s a t ( y r s )) Figure 5.
Saturation turn-off age against ( B − V ) colour. The capped errorbars correspond to the error in the fitting of the position of the saturationturn-off age with the capless error bars indicating the typical spread aboutthe turn-off age. We would expect our saturation levels to be slightly lower thanthose of P03 since they specifically select stars that are saturatedand are constrained by requiring rotation periods for the stars.Since stars will be born with some finite spread of rotation rateshowever there will be (particularly near the saturation turn-off age)some stars included in the saturated regime that are rotating moreslowly than their cluster companions. These stars will becomeunsaturated sooner than the rest of the stars in the cluster, the effectof which would be to lower the apparent saturation level slightly.Additionally rotation periods are generally easier to obtain for themost active stars.Fig. 5 shows the saturation turn-off ages obtained against ( B − V ) . Here, unlike for the saturated luminosity ratio there is noobvious trend of the saturation turn-off age with ( B − V ) . Takingthe whole FGK range the behaviour of the saturation turn-off age isconsistent with a scatter around an age of ∼
100 Myr.The gradient of the linear fit to the non-saturated regimegives us α , the index in the power law ( L X /L bol ) =( L X /L bol ) sat ( t/τ sat ) − α , where t is the age of the star and τ sat is the saturation turn-off time for that class of star. − α isthus simply the slope of the unsaturated regime. We find thatthe mean value of α is . ± . and there is no obvioustrend in the values of α either across the whole ( B − V ) range of this work or ignoring the earliest 2 bins to accountfor any possible dynamo transition. This is slightly larger thanthe α = 1 that we would expect for a t − . evolution ofrotation frequency with time coupled with an L X /L bol ∝ ω dependence first suggested by Skumanich (1972). However wenote that more recent studies, e.g. Soderblom, Duncan & Johnson(1991), Pace & Pasquini (2004), Preibisch & Feigelson (2005) findfaster rates for the rotational frequency evolution, mostly in therange t − . − t − . , which is similar to the t − . relation wefind assuming an L X /L bol ∝ ω dependence. Additionally there isa degree of degeneracy between α and the saturation turn-off age asthe adjustment to the fits of Fig. 2 that produces the minimal change to the goodness-of-fit is one which increases or decreases both thesaturation turn-off age and α simultaneously.Ideally we would include X-ray data from clusters older thanthe Hyades to better constrain the behaviour of the unsaturatedregime. X-ray studies of such clusters are few however andsuffer from low detection rates due to the lower fractional X-rayluminosities of older stars. As mentioned in Section 2.1 we includethe sample of field stars used by P03 assuming an age comparablewith solar at ∼ . Gyrs on our plots in Fig. 2 (though this sampleis not used in any of the fitting) to give a rough guide as to howreasonable it is to extend our result for the unsaturated regime toolder stars. For the most part our unsaturated regime fit passesbelow the mean of the field stars, though always within the scatterin the sample. As such we consider that, while caution should betaken, our unsaturated regime results can reasonably be extended tostars of solar age. The rather large scatter in the field star sampleis in part likely a result of the assumption of a uniform age of4.5Gyrs for the sample. Within a cluster all of the stars will havevery similar ages, albeit that this age will have an uncertainty.On the other hand for the field stars we would expect there to bea real, and probably significant, spread in age and thus that thefield stars would in fact be expected to occupy an elliptical regionaligned with the unsaturated regime. Unfortunately the ages of fieldstars is very difficult to estimate. We do however note the study ofGarc´es, Catal´an & Ribas (2011) who use wide binaries consistingof a white dwarf and a K-M type star combined with white dwarfcooling models to produce a sample of old stars with well definedages. Also note that since we include earlier type stars in our studythan that of P03 there are no field stars in the lowest bin.Overall our results for the evolution of the X-ray luminosity ofFGK stars can be summarised by the equations: L X /L bol = ( ( L X /L bol ) sat for t τ sat , ( L X /L bol ) sat ( t/τ sat ) − α for t > τ sat . (1)where ( L X /L bol ) sat , τ sat and α are taken from Table 2 for the ( B − V ) range appropriate to the star in question. Fig. 3 illustratesthese relations for each of the ( B − V ) bins we consider in thiswork. As noted in Section 2.1.3 our datasets include several types ofvariable star, particularly amongst the PMS stars. The major typesof variable star we find in our datasets, that are not excluded, are TTauri, BY Draconis and flare stars.T Tauri stars are a class of PMS star that exhibit rapid andunpredictable variability and are likely to be associated with flaring(e.g. Appenzeller & Mundt 1989). As PMS stars they will also havelarger radii than their final MS state and thus be more luminous, andas they lack a hydrogen burning core they will have deep convectivezones. We do not exclude T Tauri stars as their higher luminositywill be taken into account in the calculations and flaring eventsduring observations should be identifiable (see flare stars, below).BY Draconis variables are late type variables characterisedby periodic, rotational, brightness variations due to extensive starspot coverage and longer term modulations due to changing starspot configurations (e.g. Alekseev 2000). This variability is linkedto coronal activity of the same type that leads to X-ray emissionand as such we expect a BY Draconis designation to merely be anindicator of, likely saturated, coronal X-ray activity. Any variabilityin X-rays due to star spots rotating into and out of view is likely to c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley be fairly low level and so should not have significant effects on themeasured X-ray luminosity, though this may contribute to some ofthe observed scatter.Flare stars, also known as UV Ceti stars, exhibit short lived, butvery powerful outbursts (e.g. Tovmassian et al. 2003). These flaresare X-ray bright and can dramatically increase the X-ray luminosityof the star. Outside a flaring event however the stars are likely tohave normal saturated X-ray emission (since flaring is associatedwith strong coronal activity). As a large fraction of late type starsin young open clusters (e.g. ∼ of Pleiades K type stars) areclassified as flare stars we feel that the benefit of the greater numberstatistics is worth the risk of including an unidentified flaring event.From Jeffries et al. (2006) we expect that only a small proportionof stars that are identified as regularly undergoing flares will be inflare during any one observation. Where data exists on flares thatoccured during the observations we utilised this to remove stars thatwere definitely in flare. In addition (as indicated in Section 2.1.3)some proportion of flaring events can be identified purely by theextreme X-ray brightening of the star independent of detailed flarestudies.A substantial proportion of the stars in our sample are likelyto be binaries or multiples and clearly this will have some effect onthe X-ray to bolometric luminosity ratios obtained. The ( B − V ) colour of a binary or multiple system will be closest to that of themost massive component, since this will be the brightest optically.In the case of a binary or multiple with components of similarmass the ratio of the total X-ray to total bolometric luminosityof the system should be similar to that of the individual starsand so have minimal effect on the distributions. For systems withcomponents of rather unequal mass, e.g. a pair with types F andK, the hotter star will dominate the optical/bolometric luminositybut the X-ray to bolometric luminosity ratio of the less massive starcould be an order of magnitude higher. The difference in bolometricluminosities however is such that even in an extreme case the X-rayluminosity of the less massive star is unlikely to be greater than thatof the more massive star, and is in general unlikely to be greaterthan half that of the more massive star. While our study does notcover M-type stars we note that P03 find the saturated luminosityratio of early-mid M stars to be very similar to that of late K stars.So while P03 suggest that M stars may have significantly longersaturated periods their very low bolometric luminosities mean wedo not expect putative M star companions to have a significanteffect.We thus expect that the largest typical increase in the log( L X /L bol ) of the star/system is ∼ . − . dex due tobinarity, with an increase of up to ∼ . dex in extreme cases.Therefore binarity will not have too great an effect on the mean log( L X /L bol ) , though there may be a small systematic raising. Aswe discussed in Section 2.1.3 in contact/interacting binaries theinteraction between the stars is likely to have significant effectson the X-ray properties of the stars and while we exclude suchsystems where they have been identified there may be unidentifiedinteracting binaries in the data. The number of interacting binariesis likely to be much lower than the number of non-interactingbinaries so this should be a small effect. The Pleiades and the similarly aged NGC2516 exhibit a bimodalpopulation with respect to log( L X /L bol ) in the range . ( B − V ) < . , with an upper branch lying at the saturated level and alower branch in an unsaturated regime. Further details will be given in a subsequent paper but note that the main effect here is simplyto increase the error in the mean X-ray to bolometric luminosityratio in the Pleiades and NGC2516 over this range. It is plausiblethat this phenomenon could influence the apparent turn-off age inthe ( B − V ) range at which this bimodal behaviour is apparent,especially in the . ( B − V ) < . bin as this lies at thecentre of the bimodal region, but we note that excluding the Pleiadesand NGC2516 from the fitting in this bin makes little difference tothe turn-off age or unsaturated regime power law obtained. As indicated in Section 1 when studying the potential evaporation ofclose-orbiting exoplanets it is important to understand, and accountfor, changes in X-ray/EUV irradiation over time. As such we nowapply the results of our study of the evolution X-ray emission fromlate-type stars to the investigation of the evaporation of exoplanets.
We select transiting planets for our study as we require the planetaryradius in order to calculate the surface gravity and investigatethe evaporation rates. We selected our planet sample from theExoplanets Encyclopaedia on 13 September 2011 and take themajority of the planetary parameters required for our analysis fromthe encyclopaedia. Where other sources were consulted these arelisted in the online material along with the planet data.While we seek to use as large a sample as possible thereare a number of systems that lack some of the data required forour analysis. As such from the initial sample of 176 transitingexoplanets we remove the KOI-703 system, the latest tranche ofWASP systems (20, 42, 47, 49 and 52 through 70) and WASP-36bas many of the characteristics of the hosts and systems are not yetwell defined. We remove GJ1214b and GJ436b as these have Mtype host stars and WASP-33b as this has an A type host, and thusfall outside the scope of this study. HAT-P-31b, Kepler-5b, 6b, 15b,OGLE-TR-182b, SWEEPS-04b, 11b, TrES-5b, WASP-35b and 48bare removed as the spectral type of the host star is not well defined.Kepler-10c, 11g and 19b are removed as the mass of the planetis not well constained and we remove CoRoT-21b, HD 149026b,Kepler-7b, 8b, KOI-423b and KOI-428b as they have evolved hosts.In addition to systems where some of the necessary data islacking the improvements in transit surveys are now making theregime of super-Earth/sub-Neptune class planets accessible. Thismeans there are some low-mass, rocky, planets for which ourevaporation model, based on a hydrogen rich composition, wouldnot be appropriate. CoRoT-7b, 55 Cnc e, HD97658b, Kepler-10band planets b, c, d, e and f in the Kepler-11 system are allless massive than Uranus and Neptune. However models of theinteriors of super-Earths (e.g. Seager et al. 2007; Sotin et al. 2007;Wagner et al. 2011) show that, with the exception of Kepler-10band CoRoT-7b, all of these planets have bulk densities too low for aprimarily silicate rock composition. These studies suggest that forKepler-11b and 55 Cnc e a water-rich composition similar to thatof Ganymede with & water by mass is possible. HD97658band Kepler-11c, d, e and f however must have significant gaseousenvelopes and these are better thought of as sub-Neptunes thansuper-Earths. As such we do not include 55 Cnc e, CoRoT-7b, (cid:13) , 1–20 -ray – age relation and exoplanet evaporation Table 3.
Planet data. Most basic planetary parameters are taken from the Exoplanet Encyclopedia a with any additional references noted in column (15). Incolumns (2)–(7) we list the measured planetary properties, mass ( M P ), radius ( R P ), orbital period, semi-major axis ( a ), eccentricity and absolute spin-orbitmisalignment angle ( | λ | ). The surface gravity, mean density ( ρ ), mean orbital distance ( h a i ) , Roche lobe mass loss enhancement factor (1 /K ( ǫ )) andplanetary binding energy ( P E ) (columns (8)–(12)) are calculated directly from the basic planetary and/or host parameters. See Sections 3.2 for the form of h a i and K ( ǫ ) and 3.4.3 for P E , note that the value for
P E listed here does not include the Roche lobe correction. Full Table available online.surfacePlanet M P R P period a ecc. | λ | gravity ρ P h a i /K ( ǫ ) log( − P E ) ref.( M J ) ( R J ) (days) (AU) (deg.) (m/s) (kg/m ) (AU) (J)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)CoRoT-1 b 1.030 1.490 1.5090 0.0254 0 77 12.02 413 0.0254 2.27 36.26 fCoRoT-10 b 2.750 0.970 13.2406 0.1055 0.53 – 75.73 3997 0.1203 1.06 37.30CoRoT-11 b 2.330 1.430 2.9943 0.0436 0 – 29.52 1057 0.0436 1.38 36.99CoRoT-12 b 0.917 1.440 2.8280 0.0402 0.07 – 11.46 407 0.0403 1.62 36.18CoRoT-13 b 1.308 0.885 4.0352 0.0510 0 – 43.27 2503 0.0510 1.20 36.70 References
Kepler-10b or Kepler-11b in the main sample as our evaporationmodel is unlikely to describe these planets properly. We willhowever discuss 55 Cnc e, CoRoT-7b and Kepler-10b further lateras they have the potential to be the evaporated cores of gas giants(e.g. Jackson et al. 2010; Valencia et al. 2010), termed ‘chthonian’planets by Lecavelier des Etangs et al. (2004), and so may havebeen influenced by the same evaporation processes in the past.These three super-Earth class planets are listed separately at thebottom of Tables 3 and 4.Of the remaining planet hosts HAT-P-14, 15, OGLE-TR-10,111, 113 and TrES-3 and 4 lacked spectral type determinations buthad B − V data available; in these cases we estimated spectral typesfrom the B − V values. We expect reddening to be negligible forthe HAT and TrES planets as these are relatively nearby. This isnot the case for the OGLE planets though so for these we mayhave estimated a later spectral type than the reality. As our spectraltype/ ( B − V ) bins for the X-ray investigation are relatively wideand the bolometric correction is not rapidly varying with B − V we do not expect errors in the spectral type estimation to havelarge effects however. This then leaves us with a sample of 121planets, which represents a substantial extension to the sample usedin DW09, a result of the accelerating rate of exoplanet discoveries.A full list of the planets used and their parameters, and those oftheir host stars, can be found in the online material with extractsgiven in Tables 3 and 4 as a guide.In Fig. 6 we plot mass and surface gravity against orbitalperiod for our exoplanet sample after e.g. Mazeh et al. (2005),Southworth et al. (2007) and DW09. There is greater scatter in bothof our plots than in the original suggested correlations, particularlyto the top and right. The lower left of both distributions still displaysa notably sparse population however despite strong selection biasestowards detecting planets that would lie in these regions. Themass-period distribution in particular displays quite a sharp cut-offat the lower left. The upper right regions of both plots are also rathersparsely populated, but transit surveys select against long orbitalperiods and high surface gravities. Any residual deficit of detectionsin these regions is unlikely to be related to evaporation but may bea result of planetary migration or formation. online material web address M a ss ( M J ) s u r f a c e g r a v i t y ( m s - ) Orbital Period (days)
Figure 6.
Plots of the correlations reported by Mazeh et al. (2005) andSouthworth et al. (2007) for the sample of planets used in this work. Planetsexist to the top and right of the plot areas but these regions are sparselypopulated. The positions of the three super-Earths, 55 Cnc e, CoRoT-7b andKepler-10b, not included in the main sample, are shown in red
All exoplanets will experience evaporation as a result of Jeansescape of material in the high velocity tail of the thermal velocitydistribution. It should be noted that the temperature here that isthe determinant of the mass-loss rate is not the effective thermalemission temperature of the planet ( T eff ), but the temperature of c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley
Table 4.
Host star parameters. Most original data are taken from the Exoplanet Encyclopedia with any additional references listed in column (9). Wherea B − V colour is listed in column (3) the corresponding spectral type has been determined from the colour using the tables presented in Lang (1991).The bolometric magnitude is calculated from the V -band magnitude and a bolometric correction based on the spectral type (also using the tables of Lang1991). E X is the total X-ray energy emitted by the star over a period from formation to its present age as calculated using our X-ray characteristics for theappropriate spectral type as presented in Table 2 and described in Sections 2.2.2 and 2.2.3. Full Table available online.Star mass spectral V mag. bol. mag. distance Age log( E X ) Refs.( M ⊙ ) type (mag.) (mag.) (pc) (Gyr) (J)(1) (2) (3) (4) (5) (6) (7) (8) (9)CoRoT-1 0.95 G0V 13.6 13.42 460 4+ 38.73CoRoT-10 0.89 K1V 15.22 14.85 345 3 38.12CoRoT-11 1.27 F6V 12.94 12.8 560 2 38.9CoRoT-12 1.08 G2V 15.52 15.32 1150 6.3 38.79CoRoT-13 1.09 G0V 15.04 14.86 1310 1.6 39 References V mag. or distance information is lacking for these stars and thus E X was calculated using the bolometric luminosity of a typical star of the same spectraltype as the host (taken from the tables presented in Lang 1991). + No literature age is available for this star so an age of 4 Gyrs is assumed. the exosphere ( T ∞ ), as this is the atmospheric layer from which theescape will be taking place. T ∞ is typically much greater than T eff and for the hot Jupiters that we are considering will be determinedalmost exclusively by the incident X-ray/EUV flux. Lammer et al.(2003) showed that for exospheric temperatures likely presenton hot Jupiters the mean thermal velocities of the exosphericgas becomes comparable to the planetary escape velocity andthus that the Jeans formulation is no longer appropriate. Theysuggested instead that energy limited escape due to absorption ofX-ray/EUV radiation in the upper atmosphere is the best modelfor these hydrodynamic blow-off conditions. L09 constitutes themost recent and comprehensive study based on this evaporationmodel, however, as in earlier studies, they rely on relatively coarseestimates of stellar X-ray evolution.The model presented by Lecavelier des Etangs (2007) for theestimation of evaporation rates of exoplanets due to X-ray/EUVflux, and subsequently used by DW09, L09 and others, is shownbelow as Eq.2. This model uses the ratio of planetary bindingenergy and orbit-averaged X-ray/EUV power incident on the planetto predict the mass loss rates as: ˙ m = η L X R P GM P h a i , (2)Where ˙ m is the mass loss rate, L X is the X-ray/EUV luminosityof the star, M P and R P are the mass and radius of the planet, G isthe gravitational constant, h a i is the time averaged orbital distance(given by h a i = a (1 + e ) , with a the semi-major axis and e theeccentricity) and η is an efficiency factor. The gravitational bindingenergy used in this model assumes that gas giant planets have thedensity profile of an n = 1 polytrope.As indicated above this is an energy-limited model andthus will fail in very low X-ray/EUV irradiation regimes wherehydrodynamic blow-off conditions no longer apply and the escaperate calculation should revert to the Jeans formulation. Erkaev et al.(2007) showed that for a Jupiter mass exoplanet orbiting at & . AU the exospheric temperature may not be high enough to supporthydrodynamic blow-off. Very few of the planets in our sample lie at orbital distances this large and those that do will, as a result oftheir larger orbital distances, lie at low mass loss rates. As suchwe will assume that all of the exoplanets in our sample are in thehydrodynamic blow-off regime and simply note that the (alreadylow) mass loss estimates for planets with orbital distances & . AU may be overestimates.
This simple model assumes that a particle must be moved to infinityto escape the gravitational potential of a planet, whereas the truecriterion is that the particle must be moved to the edge of the Rochelobe (see Section 4.3 of Lecavelier des Etangs 2007) and as a resultthis model will be an underestimate of the true mass loss rate.Erkaev et al. (2007) showed that for the high mass ratio case of aplanet orbiting a host star the gravitational potential of the planet isadjusted by a factor that can be approximated as: K ( ǫ ) = 1 − ǫ + 12 ǫ , (3)thus enhancing the mass loss rate by a factor /K ( ǫ ) due tothe closeness of the Roche lobe boundary to the planet surface,where ǫ is a dimensionless parameter characterising the Roche lobeboundary distance and is given by: ǫ = R roche R P = h a i (cid:18) πρ P M ∗ (cid:19) , (4)With ρ P the planet density and M ∗ the mass of the host star.Including this correction due to Roche lobe effects the mass lossequation then becomes: ˙ m = η L X R P GM P h a i K ( ǫ ) . (5)As we approach ǫ = 1, K ( ǫ ) → , and thus the massloss rate enhancement becomes infinite since the planet is nowundergoing dynamical Roche lobe overflow and there is very littledependence on the X-ray/EUV irradiation. For the present day c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation conditions of the exoplanets in our sample the smallest values of ǫ are ∼ /K ( ǫ ) ∼ . , with the mean value being /K ( ǫ ) ∼ . .Thus we expect that Eq. 2 alone will on average underestimate thetrue mass loss rate by ∼ ∼ Determination of the efficiency factor η is a very complex and,for exoplanets for which the atmospheric composition is not wellknown, a poorly constrained task. L09 discuss ranges of possiblevalues of η and the contributions to it, settling on a value of . − . . On the other hand Ehrenreich & D´esert (2011) suggestrather higher values of η based on the present mass loss rates ofHD209458b and HD189733b. While it is highly likely that the realvalue of η is somewhat less than 1, given the poor constraints we,as far as possible, do not build a specific value of η into our study(thus implicitly using η = 1 as a base from which it is easiest tomove to different values of η ). We do however discuss the effectsof different possible values of η in our final analysis and providepredictions of mass loss histories for both η = 1 as the maximal,base, case and η = 0 . as a probable realistic value. A further issue that must be considered is evolution of theplanetary radius with time since the planetary radius comes intoEq. 2 both from the area over which the planet absorbs incomingX-ray/EUV radiation and the gravitational binding energy of theplanet. An unperturbed gas-giant will slowly contract, howeverfor close-orbiting planets there are a number of complications.The most important in the context of our study is if theplanet is undergoing significant mass loss. Baraffe et al. (2006)study evolution of planetary radii over time under various massloss regimes and initial planet compositions. Tidal effects canalso be significant for close-orbiting planets and is typicallyparametrised in terms of an additional internal energy source (e.g.Miller, Fortney & Jackson 2009). For a non-evaporating planet thiscauses contraction to stall at a point determined by the magnitudeof the tidal energy source.The application of detailed modelling of the evolution ofplanetary structure with mass loss such as that conducted byBaraffe et al. (2006) to all of the planets in our sample is beyondthe scope of this study, as is a detailed treatment of tidal heating. Toarrive at a tractable problem it is necessary to make a simplifyingassumption about the evolution of the planetary radius. As suchin this study we consider two cases; that of a constant radius(i.e. no evolution, and a constant absorbing area) and that of aconstant density with the radius thus scaling in an easily definedand analytically integrable way as the planet loses mass. Tracingpast mass loss back in time from the present state of a planetthe constant radius approximation will result in a lower predictedmass loss than the constant density approximation. This occurssince the younger, more massive, planet will be denser than it isin the present day and so more strongly bound and less easilyevaporated. Conversely when integrating mass loss forward froma fixed starting point the constant radius approximation will predicta higher degree of mass loss. These two cases in general bracket the results found by Baraffe et al. (2006) and we expect the truemass loss for the majority of the planets in our sample to liewithin the two predictions. The true mass loss will also tend to becloser to the constant radius approximation for planets losing largemass fractions and closer to the constant density approximation forplanets losing lower mass fractions.While we expect the constant radius and constant densityapproximations to bracket the true mass loss for the majorityof planets there may be deviations for planets with the highestand lowest predicted mass loss. In the late stages of evaporationin which a planet has lost a large fraction of its initial massBaraffe et al. (2006) find that it can enter a runaway mass lossregime in which the radius expands as it loses mass, acceleratingthe rate of mass loss. This runaway regime will not be replicatedin our estimates so we note that any planet that loses > − of its initial mass under our estimates may enter such a runawayregime. In addition planets with low levels of mass loss wouldlikely undergo some contraction and increase in density as they age,though somewhat less than a true Jupiter analogue due to the effectsof tidal heating and thermal (rather than X-ray/EUV) irradiation. Interms of tracing the mass loss histories of the present population ofplanets this would result in the degree of mass loss of some planetswith low predicted mass loss being slightly underestimated. The total X-ray energy emitted over the lifetime of a star, E totX , willbe the sum of the energy emitted during its saturated period and theenergy emitted while in the unsaturated regime. Using the resultsof Section 2 with a saturation turn-off age of τ sat , an unsaturatedregime power law index of − α and the present age of the star, t ,the energy emitted while in the saturated and unsaturated regimes, E satX and E unsatX respectively, is: E satX = L bol (cid:18) L X L bol (cid:19) sat τ sat , (6)and E unsatX = 1 α − L bol (cid:18) L X L bol (cid:19) sat τ sat " − (cid:18) τ sat t (cid:19) α − , (7)The total X-ray energy emitted over the lifetime of the star, E totX = E satX + E unsatX , will thus be: E totX = 1 α − L bol (cid:18) L X L bol (cid:19) sat τ sat " α − (cid:18) τ sat t (cid:19) α − , (8)Note that in these equations we implicitly assume that thebolometric luminosity of the star is constant through time andequal to the present bolometric luminosity. Early in the life ofthe star before it reaches the main sequence this is evidently nottrue and PMS stars have significantly higher luminosities than theirMS counterparts (see e.g. Stahler & Palla 2005). This will increasethe X-ray energy emitted during the saturated regime above thatpredicted by Eq. 6 above. Indeed if we take the . ( B − V ) < . bin (stars of roughly spectral type K3-6) starsfrom the very young cluster NGC6530 have a mean bolometricluminosity of ∼ . L ⊙ . This falls to around . − . L ⊙ inNGC2547 and to the expected MS values of ∼ . L ⊙ by aroundthe age of α -Persei and certainly reaching their main sequenceluminosity by the Pleiades age. As such we estimate that byneglecting bolometric luminosity evolution of PMS stars E satX asgiven by Eq.6 may be an underestimate by a factor of ∼ for c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley the later type stars in our study with a smaller deviation for theearlier type stars that reach the main sequence sooner. In additionthere is an error of a factor of ∼ ( L X /L bol ) sat and τ sat .As a result of the high level of X-ray emission during thesaturated period, and the fall off with time of X-ray emission duringthe unsaturated period, saturated emission comprises a significantfraction of the total lifetime X-ray emission of the star. For ourX-ray study this fraction is typically ∼
30 per cent. As the saturatedperiod is a fairly small fraction of the total lifetime of the systemwe can expect that the majority of evaporation will occur early inthe life of a planet. The implications of this are discussed further inSections 3.4.4 and 3.5.As mentioned in Section 3.2 the Extreme Ultraviolet (EUV)may also play a role in driving the evaporation of the atmosphereof a close orbiting gas giant, however we have only the X-rayluminosities. The EUV derives primarily from the same coronalprocesses as stellar X-ray emission and so we expect that it willfollow the same temporal variations as the X-ray emission.There are a number of complications when consideringthe EUV however. Firstly it is more difficult to quantify thestellar EUV emission since observations in this band are scarce,and made difficult by absorption in the interstellar medium.Sanz-Forcada et al. (2011) attempt to circumvent this problemusing coronal modelling to extrapolate the EUV flux from theX-ray flux of planet hosting stars that have been observed in theX-ray. They calibrate this with a small sample of nearby stars thathave been observed in the EUV. Unfortunately the vast majorityof planet-hosting stars that have been observed in the X-ray arenon-transiting systems reducing the information available about theplanets.There are also issues for the evaporation itself sinceEUV photons are more weakly penetrating than X-ray photons.Studies of the photoevaporation of protoplanetary discs (e.g.Ercolano, Clarke & Drake 2009; Owen et al. 2010) have shown thismeans that in the case of strong X-ray driven outflows the EUV doesnot penetrate far enough to drive further evaporation and instead justheats the outflow. Due to these issues we concentrate here solelyon evaporation induced by stellar X-ray emission, but note thatneglecting EUV emission may result in the underestimation of thetotal energy available to drive evaporation.
With these equations for the X-ray energy emitted by the star overtime we can integrate the mass loss (Eq. 2) to quantify the expectedmass loss of the known planets and to obtain destruction limits.Under the constant radius approximation integrating the mass lossequation back in time from the present conditions of a planet gives: m i − m t = 23 R G h a i ηE X , (9)while under the constant density approximation we obtain: m i − m t = 14 Gπρ h a i ηE X , (10)where m i is the initial mass of the planet, m t is the mass today, R and ρ are the radius and density of the planet (constant underthe respective approximations) and E X is the X-ray energy emittedby the host star over the appropriate time interval, in the case ofintegrating over the entire lifetime this will be E totX . M P /R P vs h a i − plane We can obtain ‘destruction lines’ from equations 9 and 10 by setting m t = 0 , i.e. requiring that all mass has been lost by the present day.For the constant radius approximation we obtain: m i R i = 23 1 G h a i ηE totX , (11)and for the constant density approximation: m i R i = 13 1 G h a i ηE totX , (12)using the fact that the density is constant in Eq. 12 to write it interms of the initial mass and radius and adding the subscript i tothe radius in Eq. 11 to illustrate the similarity. Both destructionlines thus give us a linear cut-off in the M P /R P vs h a i − plane assuggested by DW09 with the two differing by a factor of 2 as a resultof the different approximations. We thus find that a populationof planets significantly influenced by thermal evaporation shoulddisplay a linear cut-off in the M P /R P vs h a i − plane.In Fig. 7 we plot our sample of exoplanets in the M P /R P vs h a i − plane together with destruction limits corresponding to totalevaporation of a planet by an age of 4 Gyr under the constant densityapproximation (assuming that our relations for the unsaturatedregime are can be extended to 4 Gyr), i.e. we set t = 4 Gyrs inEq. 8. These destruction limits are not really constraints on the pastsurvival of the known planetary systems. Rather they test whetheranalogs of the presently known systems would have survived hadthey begun their evolution with the masses they have today. Forknown systems with young ages the destruction lines place someconstraints on the future survival of the system.Eqs. 11 and 12 are linearly dependent on η and thus rescalingthe destruction lines to different evaporation efficiencies is a simpletask. In Fig. 7 we show two example values, one correspondingto maximal efficiency for the constant density case and onecorresponding to a more realistic efficiency of η = 0 . (assuggested by L09) for a case intermediate between the constantdensity and constant radius regimes. While the upper lines in Fig. 7correspond to maximal efficiency for the constant density case notethat maximal efficiency for the constant radius case would leadto destruction lines a factor of two higher than the upper lines.Though our formulation for the mass loss will not apply to the threesuper-Earths 55 Cnc e, CoRoT-7b and Kepler-10b we note that theywould fall below, or very close to, both sets of destruction lines inFigs. 7.This still does not include all of the processes that will affectplanets and their degree of mass loss however. As noted abovethe X-ray energy emitted by the host star during the saturatedphase will be underestimated by a factor of up to 2, thoughthis may be at least partially counter-acted if, as suggested byL09, the evaporation efficiency is lower in very strong X-rayirradiation regimes. Kashyap, Drake & Saar (2008) suggest thatstars hosting close-orbiting gas-giant planets exhibit an excess ofX-ray emission. These findings apply to older, unsaturated, starsand so would increase the X-ray emission during the unsaturatedphase. We have also not yet discussed the impact of the Roche lobemass loss enhancement factor, /K ( ǫ ) . These three effects willall tend to push the present population of exoplanets closer to thedestruction lines in Figs. 7. The ages of some of the planets do ofcourse differ from 4 Gyrs, and we discuss the effects of varying agein Section 3.4.4, however 4 Gyrs is the mean age of those hosts withage estimates and we do not expect this to play a large role in theposition of the distribution. Additionally variations in the age of the c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation M P / R P ( k g / m ) F star G star K star -2 (au) -2 Figure 7.
Plot of mass squared over radius cubed against the inverse squareof the mean orbital distance for our exoplanet sample. We distinguishplanets with F, G and K type hosts and plot the corresponding destructionlimits for these spectral types assuming a 4 Gyr age. Solid lines correspondto the destruction limits for η = 1 under the constant density regime or η = 0 . under the constant radius regime. Dotted lines correspond to η = 0 . for a case intermediate to the constant density and constantradius regimes, or equivalently η = 0 . for the constant density regimeand η = 0 . for the constant radius regime. Note that CoRoT-9b andHD80606b lie to the right of the plot area and CoRoT-3b lies to the top ofthe plot area. known exoplanet population does not affect conclusions regardingtheir long term persistence.The slope of the destruction lines is quite a good match tothe cut-off in the distribution of the known gas-giant exoplanetsin the M P /R P vs h a i − plane, which we consider is a requiredsignature of population modification by thermal evaporation. Thisslope is an inherent feature of mass loss by thermal evaporation andis independent of most of the other parameters such as the efficiencyof the evaporation and the intensity of the stellar X-ray emission. As described above Fig. 7 does not take into account the effects ofthe proximity of the Roche lobe, which as discussed in Section 3.2.1are likely to be quite important. A useful property of the constantdensity approximation is that, as we can see from Eq. 4, theonly planetary parameter on which the Roche lobe mass lossenhancement factor, /K ( ǫ ) , depends is the density. Thus if thedensity is constant, K ( ǫ ) is also constant, making its influenceeasier to analyse. Incorporating the Roche lobe factor Eq. 10becomes: m i − m t = 14 Gπρ h a i ηK ( ǫ ) E X , (13)and the constant density destruction lines becomes: m i R i = 13 1 G h a i ηK ( ǫ ) E totX . (14) M P / R P ( k g / m ) F star G star K star(K( ) ) -1 (au) -2 Figure 8.
After Fig. 7 but now plotting ( K ( ǫ ) h a i ) − on the abscissa. Thesame destruction lines as on Fig. 7 are shown, but note that this is strictlyonly relevant to the constant density case, since K ( ǫ ) is only constantunder the constant density case. Note that the inclusion of K ( ǫ ) adjuststhe distribution to lie closer to the destruction lines. We can thus think of the effect of introducing the Roche lobe effectsas being to change the effective efficiency of the evaporation to η eff = η/K ( ǫ ) .In Fig. 8 we now adjust the abscissa of Fig. 7 to nowincorporate the Roche lobe factor. As /K ( ǫ ) > the effect of thisis broadly to move the distribution to the left. Not all of the planetsare shifted equally however since /K ( ǫ ) is different for each of theplanets, it is higher for planets with a smaller h a i and lower density.Planets that lie closer to the bottom left in Fig. 7, and thus whichwere closer to the destruction lines already, will be shifted furthest.As a result the distribution is closer to the destruction lines in Fig. 8than in Fig. 7. In addition, as we noted at the end of Section 3.4.1the slope of the destruction lines is independent of the evaporationefficiency and thus a cut-off with this slope is a required signatureof a population that has been modified by thermal evaporation.Importantly the addition of the Roche lobe factor improves thematch between the slope of the destruction lines and the cut-offin the distribution.With the introduction of the Roche lobe factor it is no longerpossible to formulate analytic destruction lines for the constantradius case since /K ( ǫ ) will increase over time as the planet losesmass and becomes less dense. In order to consider the constantradius case on Fig. 8 it is necessary to make an approximation aboutthe behaviour of the Roche lobe factor, the simplest of which is toassume that it is constant at the present day value. For the majorityof planets this is quite a good estimate. We can calculate initialmasses for planets in the constant radius regime numerically andthereby capture the true behaviour of the Roche lobe factor (themasses for the constant radius case in Table 5 are calculated inthis way). From the true initial masses we can calculate the initialvalues of /K ( ǫ ) and these are, on average, only 1.5% lower thanthe present values with the largest changes being ∼ c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley P l ane t a r y b i nd i ng ene r g y ( J ) = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . (F Xtot )* (J) = . = . = . = . = . = . Figure 9.
Plot of planetary binding energy against η ( F totX ) ∗ , a measure ofthe lifetime X-ray energy absorbed, for the known population of exoplanetsassuming η = 1 . For the majority of planets η ( F totX ) ∗ is calculatedusing the bolometric luminosity of the host star (crosses), where this is notpossible the bolometric luminosity of a typical star of the same spectral typeis used (circles). We also plot lines of constant fractional mass loss labelledwith the corresponding value of β under the constant radius approximationin blue at the left-hand edge and with the values of β for the constant densityregime in red at the bottom/right. Planets in the grey-shaded region are atrisk of entering a runaway mass loss regime. An alternative method to analyse the magnitude of evaporative massloss is to compare the gravitational potential energy of the planetswith the X-ray energy absorbed (as in e.g. Lecavelier des Etangs2007; DW09). As for Eq. 2 to estimate the binding energy of theplanets we assume they have the density profile of an n = 1 polytrope, for which the binding energy is GM /R .We also require the time integrated X-ray energy incident onthe planet. We define ( F totX ) ∗ ≡ E totX · R P / (4 h a i ) , with R P theradius of the planet today, as a measure of the lifetime X-ray energyincident on the planet. Under the constant radius approximationthis is exactly equal to the time integrated X-ray energy incidenton the planet. Under the constant density approximation the timeintegrated X-ray energy incident on the planet is not exactly thesame as ( F totX ) ∗ but rather is a function of ( F totX ) ∗ and thefractional mass loss.Using this definition for ( F totX ) ∗ and Eqs. 9 and 10 we can,on a plot of binding energy against η ( F totX ) ∗ as a measure of thelifetime X-ray energy absorbed, draw lines of constant fractionalmass loss by setting m i = m t / (1 − β ) where β is the fraction of theinitial mass that has been lost. Using a measure of the lifetime X-rayenergy absorbed also allows us to incorporate the (albeit typicallyrather small) effect of the variation of the ages of the planets in oursample from the mean of 4 Gyrs. Of the 121 planets in our sample98 (in 94 separate systems) have age estimates, for the remaining23 we assume an age of 4 Gyrs, equal to the mean age of the rest ofthe sample. Where published V magnitudes, distances and spectraltypes are available for the host star ( F totX ) ∗ is determined using thebolometric luminosity calculated for the host. Some of the host stars P l ane t a r y b i nd i ng ene r g y ( J ) (F Xtot )* (J) = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . = . Figure 10.
As Fig. 9 but including the Roche lobe reduction to the planetarybinding energy and using η = 0 . . The meaning of the crosses and circlesand values of β is as before. With respect to Fig. 9 the change in η induces ahorizontal shift in the distribution while the introduction of the Roche lobefactor induces a vertical shift in the distribution, orthogonal to that inducedby the change in η . lack information on their V magnitudes and/or distances howeverand in these cases the bolometric luminosity of a typical star of thesame spectral type (taken from the tables in Lang 1991) is used.These systems are indicated by an * in Tables 4 and 5.Fig. 9 shows such a plot of planetary binding energy against ( F totX ) ∗ , in which we use η = 1 . The lines of constant fractionalmass loss differ for the constant radius and constant densityapproximations as a result of the deviation of ( F totX ) ∗ from thetrue lifetime X-ray energy incident under the constant densityapproximation. The constant density lines take into account thevariation of the true lifetime X-ray energy incident on the planetwith the fractional mass loss and are exact under the assumption ofconstant density. In Fig. 9 we see that there is a high concentrationof planets around the β = 0 . / . (constant radius/constantdensity) line with a smaller number of systems near the β =0 . / . line. We consider that any planet lying in the grey-shadedregion would be at risk of entering a runaway mass loss regime.This would correspond to a β of over 0.9 under the constant densityapproximation and over 0.8 for any reasonable evolutionary path.In Fig. 9 we use η = 1 and ignore the Roche lobe correctionfactor to the binding energy of the planet, /K ( ǫ ) . Adjusting theefficiency will simply shift the entire distribution to the right inFig. 9 by a constant factor and has no effect on the lines of constantfractional mass loss.While the effect of the Roche lobe correction factor, K ( ǫ ) , issimply to reduce the present planetary binding energy the effect onthe mass loss history is more complex. As we noted in Section 3.4.2above K ( ǫ ) is constant under the constant density regime so inthis case the effect of introducing K ( ǫ ) is to shift the distributiondownwards. Again there is no change to the lines of constantfractional mass loss, though the downward shift will be different foreach planet. Under the constant radius regime however /K ( ǫ ) willbe smaller (closer to 1) at earlier times with the variation dependent c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation on the degree of mass loss. As described above for most planetsthis variation will be a small effect and we can provide a reasonableestimate of the impact of the Roche lobe factor by using the presentday value. As the initial value of /K ( ǫ ) will be smaller than thepresent value this means that the constant radius values of β (blue)in Fig. 10 will be overestimates. For low values of β the degree ofoverestimation will be insignificant, while for larger values of β thedegree of overestimation will be greater, but still only of the orderof a few per cent. Although as stated above we use the estimated age of theplanetary system (where available) to determine the X-ray energyabsorbed over the lifetime of the planet and the resulting masslost since formation these estimated ages are often subject to largeuncertainties (typically & ). Of the 98 planets (94 systems)with age estimates only 11 of these have ages less than 1 Gyr andonly 2 less than 0.6 Gyr. The X-ray luminosity of the host falls by ∼ orders of magnitude from its peak, saturated, value within thefirst Gyr and as a result for a system with a typical age of 4 Gyrsthe X-ray emission during the first Gyr accounts for ∼ of thetotal lifetime emission. This means that even though the ages ofthe exoplanetary systems are subject to rather large uncertaintiesa change from an age of 4 Gyrs to 1 Gyr will only reduce thefractional mass loss by ∼ . Similarly an increase from 4 Gyrsto an age of 10 Gyrs will only increase the fractional mass loss by ∼ , assuming that our X-ray emission relations can be extendedout to 10 Gyrs. Thus for ages beyond about 1 Gyr the fractionalmass loss is comparatively insensitive to the age of the planet. Asa corollary we expect that if a planet is going to be completelystripped of its envelope to leave a chthonian super-Earth this willmost likely happen within the first Gyr of its life.Within the first Gyr of the planet’s evolution, and certainlywithin the first few 100 Myrs (particularly during the saturatedperiod of the host star), the fractional mass loss will vary much morestrongly with age. As such uncertainties in age for those systemswith age estimates of < ∼ β = 0 . for η = 0 . and an age of only ∼ β mustbe higher, . , for an age of 4 Gyrs.As the study of exoplanets develops and the number of knownplanets grows it will thus become interesting to look for differencesin the mass distribution at different ages. The magnitude of anydifferences found would enable constraints to be placed on the valueof η and thus the impact of evaporation on the evolution of closeorbiting exoplanets. The discussion in Section 3.4.4 also has a bearing on the migrationof planets. The importance of the evaporation that takes placeduring the earliest stages of the life of the system means that if M P / R P ( k g / m ) (K( ) ) -1 (au) -2 s p i n - o r b i t m i s a li gn m en t ( deg . ) Figure 11.
Plot of mass squared over radius cubed against the inverse squareof the mean orbital distance for our exoplanet sample, including the Rochelobe factor, as in Fig. 8. Here the misalignment angle between the planetaryorbit and the stellar rotation axis indicated by the colour of the points.Black points indicate planets for which spin-orbit alignment data is not yetavailable. a planet spends a significant fraction of these early stages at alarger semi-major axis where evaporation is weaker this wouldsubstantially alter the total amount of mass lost by the planet. Thekey time-scale to compare against here is the saturated period of ∼
100 Myrs. In the case of standard disk migration this does notpose any problems since this must be completed on the . M P /R P – h a i − space. The distributionof planets that underwent disk migration would (assuming thatevaporation is, as we believe, an important effect) display alinear cut-off in this plane as described in Section 3.4.1. Thedistribution of planets that underwent non-disk based migrationwould be indistinguishable from that of the disk migration planetsprovided that the non-disk based migration is early. If however the c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley
Planet Mass (M J ) P l ane t s Mass today Initial mass P l ane t s Figure 12.
Histograms of the masses of our exoplanet sample as observedtoday, and initial masses as predicted by the energy-limited model under theconstant density approximation. Upper: using η = 1 . Lower: using η =0 . . The distributions change very little above 5 M J so we cut off the plotshere. non-disk migration generally happens later (at around 100 Myrsor more) the non-disk migration planets would be allowed toappear substantially below the cut-off in the distribution of diskmigration planets. An additional potential link between evaporationand migration is the possibility of asymmetric mass loss fromthe planet inducing orbital migration, as described by Bou´e et al.(2012).As a test of this we show in Fig. 11 our sample ofplanets in the M P /R P – h a i − plane coloured by their spin-orbitmisalignment angle. If misaligned systems represent those whichunderwent non-disk based migration while aligned systemsrepresent those which underwent disk based migration then thepresent (albeit limited) sub-sample of planets with measuredspin-orbit misalignment angles does not suggest any evidenceof a lower cut-off for non-disk migration planets. This impliesthat non-disk based migration also happens early in the life of aplanetary system, i.e. well within the first 100 Myrs. As the numberof planets with measured spin-orbit misalignment angles increaseswe will be enabled to better judge whether aligned and misalignedplanets obey the same cut-off. Having discussed the effects of evaporation efficiency and theRoche lobe correction to planetary binding energy we now considerthe initial masses and fractional mass loss predicted for our sampleof known exoplanets. In Fig. 12 we plot a histogram of the massestoday compared with the predicted initial masses for an evaporationefficiency of η = 1 and η = 0 . . In both cases there arenoticeable differences between the present day mass distributionand the predicted initial mass distribution, with the difference beingmore marked for the η = 1 case as would be expected. The =1 =0.25 F r a c t i ona l m a ss l o ss , Mass today (M J ) Figure 13.
Plot of the predicted fraction of initial mass lost against presentday mass for the planets in our sample. The horizontal dashes indicate thefractional mass loss predicted under the constant density regime, while thebottom of the vertical tail indicates the fractional mass loss predicted underthe constant radius regime. At lower fractional mass loss the differencebetween the two regimes is negligible such that the vertical tail may notbe discernible, while at higher fractional mass loss the regimes diverge. difference between the present day and initial mass distributionis negligible for higher mass planets ( & M J ), which again isas would be expected. The constant radius approximation, whichpredicts lower fractional mass loss, produces a slightly smaller shiftin the mass distribution but the differences are not dramatic.In Table 5 (full Table available online ) we give for each planetthe predicted initial mass and fractional mass loss under both theconstant density and constant radius approximation regimes for η = 1 and η = 0 . . We plot the predicted fraction of initial masslost against present day mass in Fig. 13. Planets near the upperenvelope of the distribution represent some of the most closelyorbiting examples of planets of their mass and so are, for theirmass, some of the easiest to detect given the biases inherent inplanet surveys. As such while additional planets may be discoveredin future that lie at higher fractional mass losses we do not expectthe distribution to change dramatically, in particular with planetsthat have present day masses & M J not having been subject tohigh mass loss. Similarly biases in planet surveys are the reason foran apparent lack of planets with low present day masses and lowfractional mass loss, since such planets would be further from theirhost stars and so would be more difficult to detect.The planets with the highest fractional mass loss areHAT-P-32b and WASP-12b with both predicted to have lost ∼ η = 0 . ,rising to ∼ η = 1 . WASP-12b has one of the shortestorbital periods in our sample at only 26 hours while HAT-P-32b isone of the least dense planets in our sample at 148 kg m − . Havingpresent masses in the region of 1 M J the β ’s of these planets alsotranslate into very large absolute mass loss with WASP-12b andHAT-P-32b both having lost at least 0.2 M J since formation. online material web address c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation Table 5.
Predictions for mass lost by the planets in our sample. We list the present mass of the planet ( M P ) in column (2) and then, for the different regimesdiscussed in the text, the initial masses in columns (3)–(6) with the fraction of the initial mass lost ( β ) to reach the present day mass from the initial massin columns (7)–(10). All masses are measured in Jupiter masses ( M J ) . Columns (3), (4), (7) and (8) correspond to η = 1 while columns (5), (6), (9) and(10) correspond to η = 0 . . The constant radius approximation is used for columns (3), (5), (7) and (9) while the constant density approximation is usedfor columns (4), (6), (8) and (10). See Sections 3.4.3 and 3.6 for further detail. The full Table is available online.initial mass, m i fraction of initial mass lost, β Planet M P η = 1 η = 0 . η = 1 η = 0 . cons. R cons. ρ cons. R cons. ρ cons. R cons. ρ cons. R cons. ρ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)CoRoT-1 b 1.030 1.4333 1.5456 1.1491 1.1589 0.2814 0.3336 0.1036 0.1112CoRoT-10 b 2.750 2.7503 2.7503 2.7501 2.7501 1E-4 1E-4 2E-5 2E-5CoRoT-11 b 2.330 2.3904 2.3917 2.3453 2.3454 0.0253 0.0258 0.0065 0.0066CoRoT-12 b 0.917 1.0720 1.0880 0.9586 0.9598 0.1446 0.1572 0.0434 0.0445CoRoT-13 b 1.308 1.3287 1.3288 1.3132 1.3132 0.0155 0.0157 0.0040 0.0040* V mag. or distance information is lacking for the hosts of these planets and thus the predicted fractional mass losses were calculated using the bolometricluminosity of a typical star of the same spectral type as the host (taken from the tables presented in Lang 1991). We can use Eq. 13 to estimate the minimum initial mass for a planetof a given initial (and constant) density to survive for 4 Gyrs ata given orbital distance around a star of a given spectral type bysetting m t = 0 in the same way as we formulated destruction limitsin Section 3.4.1. In general for a planet undergoing total evaporationthe density and K ( ǫ ) will decrease with time however, and as aresult this will underestimate the minimum mass. In addition whenhigh fractional mass losses are reached there is a possibility ofentering a runaway mass loss regime (as discussed earlier), andsome fraction of the planet mass may be in the form of a rockycore. To account for the possibility of runaway mass loss and aremnant core rather than setting m t = 0 we choose instead to set m t = 0 . m i in Eq. 13. In this way we obtain the minimum initialmass for survival for 4 Gyrs, m S , as: m S = 518 1 Gπρ h a i ηK ( ǫ ) E totX (15)In Fig. 14 we present estimates of the minimum initial massfor survival to 4 Gyrs for different spectral types for both η = 1 and η = 0 . and a selection of representative densities that coversmost of the range found for the planets in our sample with masses . M J . As m S is linearly dependent on η this is easily modifiedto any desired value of η . The highest density used is similar to thatof Jupiter (1326 kg m − ) while 400 kg m − is roughly the peak ofthe density distribution for our sample and 200 kg m − is typicalof the lowest density exoplanets, the lowest density present in oursample being 82 kg m − for WASP-17b.From these figures we can see that planets of the lowestdensities found in our sample are essentially precluded from beingfound at the smallest orbital distances around any spectral typeof star. We also do not expect gas giants of much less than aJupiter mass to survive at orbital distances of . . AU aroundF and G type stars unless they are quite dense. Less massive gasgiants might survive at such close orbital distances around K typestars even with more typical densities however. In comparing thepresently known exoplanets with the lines of minimum initial massfor survival we see that for high evaporation efficiencies a numberof the presently known exoplanets lie close to or below the survivallines and so we would expect a planet that had been born withthe same parameters not to have survived to the present day. Asan example for efficiencies η ≈ a planet with an initial mass of ∼ M J and a typical density of ∼
400 kg m − at an orbitaldistance of ∼ As we have mentioned earlier in Section 3.1 the energy-limitedevaporation model that we apply to the study of the evaporationof hot-Jupiters is intended for planets with a substantial hydrogencontent. It is unlikely to be suitable for super-Earths of a bulkrock composition as a number of complications become potentiallysignificant such as the latent heat of sublimation and high ionisationstates of the large fraction of heavy elements. Nonetheless arocky or rock/water composition is not enough to render planetsin extremely close orbits such as CoRoT-7b, Kepler-10b and 55Cnc e immune to evaporation since the stellar irradiation is likelysufficient to melt regions of the crust and even produce a tenuousatmosphere of vapourised silicate minerals (e.g. Schaefer & Fegley2009, L´eger et al. 2011). As such the energy limited model can stillbe used to provide upper limits on the mass loss of super-Earth classplanets. At an evaporation efficiency of η = 0 . all 3 super-Earthswould have lost more than 20 per cent of their mass since formationif they have always had roughly their present compositions. For 55Cnc e this would make its initial mass comparable to that of Uranus.Even for an evaporation efficiency of 0.1 all three would have lostmore than half an Earth mass.This suggests that even a planet that has always had apredominantly rocky composition can be significantly affected byevaporation. In addition the densities of the super-Earths, beingan order of magnitude higher than those typical of close orbitinggas giants, make their mass loss estimates far lower than wouldbe the case for a gas giant under the same conditions. This raisesthe question, as also discussed by e.g. Jackson et al. (2010) andValencia et al. (2010), of whether it is possible that these very closeorbiting super-Earths could be chthonian planets – the remnantcores of gas or ice giants that have been completely stripped oftheir atmospheres. Using Fig. 14 we see that for a K type host a c (cid:13) , 1–20 A.P. Jackson, T.A. Davis and P.J. Wheatley M a ss t oda y ( M J ) m i n i m u m s u r v i v a l m a ss ( M J ) orbital distance (AU) 200 kg m -3
400 kg m -3 -3 FGK M a ss t oda y ( M J ) m i n i m u m s u r v i v a l m a ss ( M J ) orbital distance (AU) 200 kg m -3
400 kg m -3 -3 FGK
Figure 14.
Plots of the estimated minimum initial mass for a planet to survive for 4 Gyrs for a selection of different planet densities and host spectral typesas a function of orbital distance; left for η = 1 , right for η = 0 . . Included for comparison are the present day parameters of planets from our sample inbands around the densities used for the curves ( <
300 kg m − , 300-500 kg m − , 1000-1400 kg m − ). The colouration of the lines and points indicate thedensity band to which they correspond while line and point style corresponds to the different spectral types. gaseous planet of . . M J at the orbital distance of CoRoT-7b or55 Cnc e and a typical density of 400 kg m − would be completelyevaporated by 4 Gyrs. For a G type host a similar planet of . . M J would not survive to an age of 4 Gyrs at the orbital distanceof Kepler-10b. It thus seems possible that these three planets couldbe chthonian planets though they may also have been born rocky. We have used archival X-ray surveys of open clusters to study thecoronal X-ray activity-age relationship in late-type stars (in therange . ( B − V ) < . ), with a particular focus onconstraining the regime of saturated X-ray emission. We find a trendfor a decrease in the saturated value of the X-ray to bolometricluminosity ratio across this ( B − V ) range from − . to − . for the latest to earliest type stars in our study. Thesaturation regime turn-off ages across the ( B − V ) range of ourstudy are consistent with a scatter around an age of ∼
100 Myrs.We also point the reader towards the complementary study of theactivity-rotation relation by Wright et al. (2011) that was publishedwhile this work was in the refereeing process.In the unsaturated regime we find that the mean value of α in the power law ( L X /L bol ) = ( L X /L bol ) sat ( t/τ sat ) − α is . ± . with no obvious trend with spectral type. Under theassumption of an L X /L bol ∝ ω dependence this corresponds toan ω ∝ t − . evolution of rotational frequency with time. Theresults of our X-ray study are summarised by Eq. 1 and Table 2. Wenote that while our unsaturated regime power laws seem broadlyconsistent with the field star sample of P03 our results should onlybe extended to stars older than the Hyades with caution since theseare the oldest stars in our cluster sample. Future deep X-ray surveysof old open clusters, or better age estimates for field stars, wouldenable the evolution of X-ray luminosity in the unsaturated regimeto be more strongly constrained.We have applied our improved constraints on the evolution ofthe X-ray luminosity of late-type stars to evaporational evolutionof close-orbiting exoplanets using the energy-limited model ofLecavelier des Etangs (2007) and including a more accurate description of Roche lobe effects as described by Erkaev et al.(2007). With a substantially larger sample of planets we confirmthe finding of DW09 that the planet distribution displays a linearcut-off in the M P /R P vs h a i − plane. We also confirm that sucha cut-off is an expected feature of modification of the populationby thermal evaporation irrespective of efficiency for any valuestypically considered.We provide estimates of the past thermal mass loss of theknown transiting exoplanets, finding that in the case of a constantevaporation efficiency of η = 0 . , 11/121 planets (10 per cent)have lost more than 5 per cent of their mass since formation. Inthe case of highly efficient evaporation this fraction rises to a third( ∼ < Gyr,with the most marked changes occurring in the first 100-200 Myr,during the saturated phase of the host. With the increasing rate ofexoplanet discoveries and improvements in detection it should soonbe possible to test this prediction by comparing planet distributionsin young clusters of different ages.The importance of the earliest phases of evaporation also hasimplications for planetary migration. Late migration ( & Myrs)would spare planets from the some of the worst effects ofevaporation leading to significantly lower lifetime mass loss. Aswork continues to be done on the migration of exoplanets in to closeorbits, for example via measurements of the Rossiter-McLaughlineffect, it will be possible to divide the exoplanets into populationsthat underwent disk migration or non-disk migration. Differences,or not, between the distributions in the M P /R P vs h a i − plane c (cid:13) , 1–20 -ray – age relation and exoplanet evaporation of planets that arrived at their present orbital positions throughdifferent migration mechanisms will allow constraints to be placedon the epoch at which migration occurs. The authors would like to thank the anonymous referee forcomments which were helpful in refining this manuscript. AJis supported by an STFC Postgraduate Studentship, PW issupported by an STFC rolling grant. This work has made useof the valuable resources of the SIMBAD database and VizieRcatalogue access tool, operated at CDS, Strasbourg, France andthe Exoplanet Encyclopaedia, maintained by J. Schneider at theParis Observatory. The research leading to these results has receivedfunding from the European Community’s Seventh FrameworkProgramme (/FP7/2007-2013/) under grant agreement No 229517.
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