Characterizing Distant Worlds with Asterodensity Profiling
MMon. Not. R. Astron. Soc. , 1–26 (2013) Printed 1 November 2018 (MN L A TEX style file v2.2)
Characterizing Distant Worlds with AsterodensityProfiling
David M. Kipping , (cid:63)(cid:63) Harvard-Smithsonian Center for Astrophysics, 60, Garden Street, Cambridge, MA 02138 Carl Sagan Fellow
Accepted 2014 February 12. Received 2014 January 10; in original form 2013 November 4
ABSTRACT
Eclipsing systems, such as transiting exoplanets, allow one to measure themean stellar density of the host star under various idealized assumptions. Aster-odensity Profiling (AP) compares this density to an independently determinedvalue in order to check the validity of the assumptions and ultimately deriveuseful parameters. Several physical effects can cause said assumptions to becomeinvalid, with the most well-known example being the so-called photo-eccentriceffect. In this work, we provide analytic expressions for five other effects whichinduce AP deviations: the photo-blend, -spot, -timing, -duration and -mass ef-fects. We find that these effects can easily reproduce large AP deviations andso we caution that extracting the eccentricity distribution is only viable withcareful consideration of the prior distributions for these other effects. We alsore-investigate the photo-eccentric effect and derive a single-domain minimumeccentricity expression and the parameter range for which analytic formulaeare valid. This latter result shows that the assumptions underlying the analyticmodel for the photo-eccentric effect break down for close-in, highly-eccentricplanets, meaning that extreme care must be taken in this regime. Finally, wedemonstrate that contaminated light fraction can be solved for, indicating thatAP could be a potent tool for planet validation.
Key words: techniques: photometric — methods: analytical — asteroseismol-ogy — planet and satellites: fundamental parameters — eclipses
Asterodensity profiling (AP) is a relatively new conceptin the study of astronomical eclipses, such as transit-ing planets and eclipsing binaries, with the potential toconstrain various properties of an eclipsing system usingphotometric data alone. AP exploits a well-known trickin the field of photometric eclipses that if an object tran-sits across the face of a star multiple times, then one canmeasure the mean density of the host star, ρ (cid:63) , using Ke-pler’s Third Law alone, under various idealized assump-tions. This was first demonstrated in the pioneering workof Seager & Mall´en-Ornelas (2003) and the most commonapplication of this trick in the study of exoplanets hasbeen to use the ρ (cid:63) measurement as a luminosity indicator (cid:63) E-mail: [email protected] for stellar evolution models, in order to obtain physicaldimensions for the host star (Sozzetti et al. 2007).AP goes further than this though, by comparing thetransit light curve derived stellar density, ρ (cid:63), obs , to someindependent measure of the same term, ρ (cid:63), true , in orderto test the validity of the idealized assumptions and ul-timately extract information on the state the eclipsingsystem. If all of the idealized assumptions made in thedefinition of ρ (cid:63), obs are correct (Seager & Mall´en-Ornelas2003), then naturally one expects ( ρ (cid:63), obs /ρ (cid:63), true ) = 1 (towithin the measurement uncertainties). Any deviationfrom unity implies that one or more of the idealized as-sumptions are invalid and the magnitude and directionof this deviation provide insights into the physical originof the discrepancy. These idealized assumptions include(but are not limited to) an opaque planet, a sphericalplanet, a spherical star, non-variable transit shape, Ke- c (cid:13) a r X i v : . [ a s t r o - ph . E P ] J un David M. Kipping plerian circular orbit and negligible blending from unre-solved luminous objects.The first usage of the term “asterodensity profiling”was by Kipping et al. (2012), who focussed on Multi-body Asterodensity Profiling (MAP) to constrain mutualorbital eccentricities. By focussing on systems with mul-tiple transiting planets, several measurements of ρ (cid:63), obs are obtained, allowing one to seek relative discrepan-cies in ρ (cid:63), obs , rather than the absolute discrepancy deter-mined when ρ (cid:63), true is known. MAP is particularly pow-erful since it makes no assumption about the true stellardensity.Although Kipping et al. (2012) briefly speculatedthat Single-body Asterodensity Profiling (SAP) wouldbe plausible if a very tight constraint on ρ (cid:63), true wasavailable, such as that from asteroseismology, Dawson& Johnson (2012) proposed that even a loose prioron ρ (cid:63), true would be sufficient to identify highly ec-centric planets. Referring to the effect as the “photo-eccentric effect”, the authors demonstrated the techniqueon the known eccentric planet HD 17156b obtaining e = 0 . +0 . − . in good agreement with the radial veloc-ity determination of e = 0 . ± .
08. In later work, thesame authors showed that the
Kepler planetary candi-date KOI-1474.01 has an eccentricity of e = 0 . +0 . − . ,if the candidate is genuine (Dawson et al. 2012). In thecase of ostensibly near-circular orbits, SAP provides lessconstraining determinations; for example Kipping et al.(2013) recently used SAP on Kepler-22b to determine e = 0 . +0 . − . (we propose an explanation for this in § ρ (cid:63), obs is a function ofthe observed transit durations (as shown later in § ρ (cid:63), true is a directobservable from the “gold standard” inference from as-teroseismology, using the frequency spacing of pulsationsmodes (Ulrich 1986). In contrast, the “true” transit du-ration or maximum transit duration can, in general, onlybe inferred by invoking stellar evolution models since oneneeds to estimate the stellar radius, R (cid:63) (Moorhead et al.2011).AP, and variants thereof, have so far been predomi-nantly employed for constraining orbital eccentricities inboth individual systems (e.g. Dawson & Johnson 2012;Dawson et al. 2012; Kipping et al. 2013) and with re-gard to the entire eccentricity distribution (e.g. Moor-head et al. 2011; Kane et al. 2012; Dawson et al. 2013).The former goal has a particularly important place withregard to assessing habitability of planetary candidatessince eccentricity can have severe effects (Dressing et al.2010). The latter is mostly concerned with testing planet-formation models (Ford & Rasio 2008; Juri´c & Tremaine2008; Socrates et al. 2012; Dong et al. 2013).The importance of measuring eccentricities is there- fore apparent; thus explaining the recent focus of ap-plying AP for constraining eccentricities via the photo-eccentric effect. However, relatively little work exists inthe literature exploring the other physical effects whichcan lead to ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1. This absence of inves-tigation is problematic since a circular orbit is not theonly idealized assumption in the definition of ρ (cid:63), obs whichmay be in error, and thus responsible for an observa-tion that ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1. Critically, negating theseother effects may lead to systematic errors in derivedeccentricities or even completely erroneous conclusionsabout the state of a system. Additionally, the analyticexpressions for AP are only approximate forms (Kip-ping et al. 2012), and yet the explicit valid range fortheir applicability remains unknown. The purpose of thiswork is to provide analytic expressions for several plau-sible alternative mechanisms by which AP can produce( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1 and define the exact parameter rangefor which these expressions may be reasonably employedwithout a significant loss of accuracy. We therefore aimto provide a foundational theoretical framework for thisburgeoning field of study. ρ (cid:63), obs It is not the purpose of this work to provide a detailedintroductory review of basic transit theory. Despite this,we here provide a brief synposis of how the mean stellardensity is derived in the context of the AP technique.Those interested in a more detailed pedagogical discus-sion are directed to Winn (2010).Throughout this work, including all appendices, wemake the fundamental assumption that any observedtransits satisfy the criteria 0 < b < (1 − p ) where b is the impact parameter of the transiting object of p isthe ratio-of-radii between the transiting object and thehost star. In the absence of limb-darkening, such a tran-sit would be described as exhibiting a flat-bottom. Since b > p < (cid:4) δ obs : the observed transit depth (cid:4) τ obs : the observed time of transit minimum (cid:4) T , obs : the observed first-to-fourth contact transitduration (cid:4) T , obs : the observed second-to-third contact transitdurationThe transit depth scales with the size of the tran-siting object and thus p obs is easily recovered. Multipleepochs provide several τ obs measurements which can beused to infer the orbital period of the transiting object, c (cid:13) , 1–26 sterodensity Profiling P . The other two observables, T , obs and T , obs , may beused to determine the observed impact parameter, b obs ,and the observed scaled semi-major axis of the orbit,( a/R (cid:63) ) obs , as demonstrated by Seager & Mall´en-Ornelas(2003). Under the assumption of a spherical, opaque,dark planet on a Keplerian circular orbit transiting aspherical, unblended host star, Seager & Mall´en-Ornelas(2003) showed that the transit durations would be givenby T = Pπ sin − (cid:34)(cid:115) (1 ± p ) − b ( a/R (cid:63) ) − b (cid:35) . (1)These expressions may be solved simultaneously for b and ( a/R (cid:63) ). We refer to these expressions as the ob-served impact parameter and observed scaled semi-majoraxis since both terms are only valid under the various as-sumptions made thus far. b ≡ (1 − p obs ) − sin ( T , obs π/P )sin ( T , obs π/P ) (1 + p obs ) − sin ( T , obs π/P )sin ( T , obs π/P ) (2)( a/R (cid:63) ) ≡ (1 + p obs ) − b [1 − sin ( T , obs π/P )]sin ( T , obs π/P ) . (3)It is now trivial to show that ρ (cid:63), obs is found usingKepler’s Third Law: ρ (cid:63), obs ≡ π ( a/R (cid:63) ) GP , (4)where G is the Gravitational constant. Equation 4also assumes M P (cid:28) M (cid:63) in addition to the previous as-sumptions and we use the equivalent symbol since theabove represents a definition which we will use through-out this work. Note, that we refer to this density withthe subscript “obs” for observed, whereas previous workshave used the subscript “circ” for circular (e.g. Dawson& Johnson 2012; Kipping et al. 2012). The reason forthis change is that, as demonstrated throughout this pa-per, numerous other idealized assumptions are made toderive Equation 4 in addition to a circular orbit and it issomewhat misleading to label the term with “circ” sinceit implies that this is the only relevant assumption.It is important to stress that limb darkening pa-rameters do not feature in the calculation of ρ (cid:63), obs . Inother words, ρ (cid:63), obs is not functionally dependent uponthe limb darkening coefficients (LDCs) or profile; e.g. ρ (cid:63), obs (cid:54) = f ( u , u ) in the case of quadratic limb dark-ening. This can be understood on the basis that theLDCs do not affect the instant at which the planet’sprojected disc contacts the star’s projected disc i.e. thecontact points, since this is purely dynamical. Therefore,the transit durations, T , obs and T , obs , are not affectedby the LDCs in anyway. Since ρ (cid:63), obs depends solely upon p obs , P , T , obs & T , obs , then ρ (cid:63), obs must also be inde-pendent of the LDCs. In practice, one could arrive at the wrong ρ (cid:63), obs by fixing the LDCs to some values which donot represent the truth. This would lead to a biased esti-mate of p obs , T , obs & T , obs , and consequently a biasedestimate of ρ (cid:63), obs . We therefore advocate careful selectionof the priors in the LDCs and specifically suggest em-ploying the non-informative prior basis set proposed inKipping (2013), which will propagate the uncertainty ofthe LDCs into the derivation of ρ (cid:63), obs . Essentially, thismeans that the derived ρ (cid:63), obs value loses precision butgains accuracy - a satisfactory compromise in most cases.Having established that it makes no difference to any ofthe derivations in this work whether we include/excludelimb darkening, many of the figures in this paper willnegate it for the sake of clarity but once again we stressthat it does not affect the validity of the derived expres-sions.Finally, we note that the reason why we earlierstated that we will assume b < (1 − p ) at all times isevident from the above expressions, since T is unde-fined otherwise and thus it is not possible to calculate ρ (cid:63), obs . Therefore, using the approach of Seager & Mall´en-Ornelas (2003), one can only measure the light curve de-rived stellar density of a star if b < (1 − p ) and thus APis only possible in such a regime. Ideally, the observed transit depth and durations areequivalent to the true values. In such a case, one shouldexpect (to within the measurement uncertainties) thatlim idealized assumptions valid (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = 1 . However, as is shown in this work, there are manyrealistic conditions which do not satisfy the ideal transitassumptions made in Seager & Mall´en-Ornelas (2003).Rather than seeing this as nuisance though, the princi-ple of AP is to exploit the ( ρ (cid:63), obs /ρ (cid:63), true ) ratio to notonly test the validity of the idealized assumptions but toactually infer properties of an eclipsing system by anal-ysis of the magnitude and direction of any discrepancies(or lack there-of).As mentioned earlier, either an independent measureof ρ (cid:63), true is required to perform the SAP variant or rel-ative differences between multiple transiting object canbe used to perform MAP. There are many different physical scenarios which cancause a significant AP discrepancy (which we defineas when ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1 at high significance). Inthis work, we attempt to derive analytic expressions forseveral important effects to aid observers interpretingsuch measurements. In general, an unaccounted for effect(dubbed a “photo- name effect” throughout this work) c (cid:13)000
Kepler planetary candi-date KOI-1474.01 has an eccentricity of e = 0 . +0 . − . ,if the candidate is genuine (Dawson et al. 2012). In thecase of ostensibly near-circular orbits, SAP provides lessconstraining determinations; for example Kipping et al.(2013) recently used SAP on Kepler-22b to determine e = 0 . +0 . − . (we propose an explanation for this in § ρ (cid:63), obs is a function ofthe observed transit durations (as shown later in § ρ (cid:63), true is a directobservable from the “gold standard” inference from as-teroseismology, using the frequency spacing of pulsationsmodes (Ulrich 1986). In contrast, the “true” transit du-ration or maximum transit duration can, in general, onlybe inferred by invoking stellar evolution models since oneneeds to estimate the stellar radius, R (cid:63) (Moorhead et al.2011).AP, and variants thereof, have so far been predomi-nantly employed for constraining orbital eccentricities inboth individual systems (e.g. Dawson & Johnson 2012;Dawson et al. 2012; Kipping et al. 2013) and with re-gard to the entire eccentricity distribution (e.g. Moor-head et al. 2011; Kane et al. 2012; Dawson et al. 2013).The former goal has a particularly important place withregard to assessing habitability of planetary candidatessince eccentricity can have severe effects (Dressing et al.2010). The latter is mostly concerned with testing planet-formation models (Ford & Rasio 2008; Juri´c & Tremaine2008; Socrates et al. 2012; Dong et al. 2013).The importance of measuring eccentricities is there- fore apparent; thus explaining the recent focus of ap-plying AP for constraining eccentricities via the photo-eccentric effect. However, relatively little work exists inthe literature exploring the other physical effects whichcan lead to ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1. This absence of inves-tigation is problematic since a circular orbit is not theonly idealized assumption in the definition of ρ (cid:63), obs whichmay be in error, and thus responsible for an observa-tion that ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1. Critically, negating theseother effects may lead to systematic errors in derivedeccentricities or even completely erroneous conclusionsabout the state of a system. Additionally, the analyticexpressions for AP are only approximate forms (Kip-ping et al. 2012), and yet the explicit valid range fortheir applicability remains unknown. The purpose of thiswork is to provide analytic expressions for several plau-sible alternative mechanisms by which AP can produce( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1 and define the exact parameter rangefor which these expressions may be reasonably employedwithout a significant loss of accuracy. We therefore aimto provide a foundational theoretical framework for thisburgeoning field of study. ρ (cid:63), obs It is not the purpose of this work to provide a detailedintroductory review of basic transit theory. Despite this,we here provide a brief synposis of how the mean stellardensity is derived in the context of the AP technique.Those interested in a more detailed pedagogical discus-sion are directed to Winn (2010).Throughout this work, including all appendices, wemake the fundamental assumption that any observedtransits satisfy the criteria 0 < b < (1 − p ) where b is the impact parameter of the transiting object of p isthe ratio-of-radii between the transiting object and thehost star. In the absence of limb-darkening, such a tran-sit would be described as exhibiting a flat-bottom. Since b > p < (cid:4) δ obs : the observed transit depth (cid:4) τ obs : the observed time of transit minimum (cid:4) T , obs : the observed first-to-fourth contact transitduration (cid:4) T , obs : the observed second-to-third contact transitdurationThe transit depth scales with the size of the tran-siting object and thus p obs is easily recovered. Multipleepochs provide several τ obs measurements which can beused to infer the orbital period of the transiting object, c (cid:13) , 1–26 sterodensity Profiling P . The other two observables, T , obs and T , obs , may beused to determine the observed impact parameter, b obs ,and the observed scaled semi-major axis of the orbit,( a/R (cid:63) ) obs , as demonstrated by Seager & Mall´en-Ornelas(2003). Under the assumption of a spherical, opaque,dark planet on a Keplerian circular orbit transiting aspherical, unblended host star, Seager & Mall´en-Ornelas(2003) showed that the transit durations would be givenby T = Pπ sin − (cid:34)(cid:115) (1 ± p ) − b ( a/R (cid:63) ) − b (cid:35) . (1)These expressions may be solved simultaneously for b and ( a/R (cid:63) ). We refer to these expressions as the ob-served impact parameter and observed scaled semi-majoraxis since both terms are only valid under the various as-sumptions made thus far. b ≡ (1 − p obs ) − sin ( T , obs π/P )sin ( T , obs π/P ) (1 + p obs ) − sin ( T , obs π/P )sin ( T , obs π/P ) (2)( a/R (cid:63) ) ≡ (1 + p obs ) − b [1 − sin ( T , obs π/P )]sin ( T , obs π/P ) . (3)It is now trivial to show that ρ (cid:63), obs is found usingKepler’s Third Law: ρ (cid:63), obs ≡ π ( a/R (cid:63) ) GP , (4)where G is the Gravitational constant. Equation 4also assumes M P (cid:28) M (cid:63) in addition to the previous as-sumptions and we use the equivalent symbol since theabove represents a definition which we will use through-out this work. Note, that we refer to this density withthe subscript “obs” for observed, whereas previous workshave used the subscript “circ” for circular (e.g. Dawson& Johnson 2012; Kipping et al. 2012). The reason forthis change is that, as demonstrated throughout this pa-per, numerous other idealized assumptions are made toderive Equation 4 in addition to a circular orbit and it issomewhat misleading to label the term with “circ” sinceit implies that this is the only relevant assumption.It is important to stress that limb darkening pa-rameters do not feature in the calculation of ρ (cid:63), obs . Inother words, ρ (cid:63), obs is not functionally dependent uponthe limb darkening coefficients (LDCs) or profile; e.g. ρ (cid:63), obs (cid:54) = f ( u , u ) in the case of quadratic limb dark-ening. This can be understood on the basis that theLDCs do not affect the instant at which the planet’sprojected disc contacts the star’s projected disc i.e. thecontact points, since this is purely dynamical. Therefore,the transit durations, T , obs and T , obs , are not affectedby the LDCs in anyway. Since ρ (cid:63), obs depends solely upon p obs , P , T , obs & T , obs , then ρ (cid:63), obs must also be inde-pendent of the LDCs. In practice, one could arrive at the wrong ρ (cid:63), obs by fixing the LDCs to some values which donot represent the truth. This would lead to a biased esti-mate of p obs , T , obs & T , obs , and consequently a biasedestimate of ρ (cid:63), obs . We therefore advocate careful selectionof the priors in the LDCs and specifically suggest em-ploying the non-informative prior basis set proposed inKipping (2013), which will propagate the uncertainty ofthe LDCs into the derivation of ρ (cid:63), obs . Essentially, thismeans that the derived ρ (cid:63), obs value loses precision butgains accuracy - a satisfactory compromise in most cases.Having established that it makes no difference to any ofthe derivations in this work whether we include/excludelimb darkening, many of the figures in this paper willnegate it for the sake of clarity but once again we stressthat it does not affect the validity of the derived expres-sions.Finally, we note that the reason why we earlierstated that we will assume b < (1 − p ) at all times isevident from the above expressions, since T is unde-fined otherwise and thus it is not possible to calculate ρ (cid:63), obs . Therefore, using the approach of Seager & Mall´en-Ornelas (2003), one can only measure the light curve de-rived stellar density of a star if b < (1 − p ) and thus APis only possible in such a regime. Ideally, the observed transit depth and durations areequivalent to the true values. In such a case, one shouldexpect (to within the measurement uncertainties) thatlim idealized assumptions valid (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = 1 . However, as is shown in this work, there are manyrealistic conditions which do not satisfy the ideal transitassumptions made in Seager & Mall´en-Ornelas (2003).Rather than seeing this as nuisance though, the princi-ple of AP is to exploit the ( ρ (cid:63), obs /ρ (cid:63), true ) ratio to notonly test the validity of the idealized assumptions but toactually infer properties of an eclipsing system by anal-ysis of the magnitude and direction of any discrepancies(or lack there-of).As mentioned earlier, either an independent measureof ρ (cid:63), true is required to perform the SAP variant or rel-ative differences between multiple transiting object canbe used to perform MAP. There are many different physical scenarios which cancause a significant AP discrepancy (which we defineas when ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) = 1 at high significance). Inthis work, we attempt to derive analytic expressions forseveral important effects to aid observers interpretingsuch measurements. In general, an unaccounted for effect(dubbed a “photo- name effect” throughout this work) c (cid:13)000 , 1–26 David M. Kipping will cause a systematic and constant deviation in ei-ther the depth or the duration such that p obs (cid:54) = p true , T , obs (cid:54) = T , true and/or T , obs (cid:54) = T , true . Unaccounted-for periodic transit timing/duration/depth variations(TTV/TDV/T δ V) induced by perturbing gravitationalinfluences or starspots can be interpreted as a system-atic, constant deviation in the composite transit lightcurve’s durations and/or depth too, as shown later in § § § ρ (cid:63), obs by considering it to be functionally dependent via: ρ (cid:63), obs [ p obs ( p true , X ) , T , obs ( T , true , X ) , T , obs ( T , true , X )] , where X is a vector of arbitrary length representingthe parameters which describe the unaccounted-for phys-ical effect(s). In practice, one computes the expressionsfor p obs , T , obs and T , obs and then uses Equations 2, 3& 4 to analytically express ρ (cid:63), obs ( p true , b true , ρ (cid:63), true ). Inpractice, the the derived expression is often extremelycumbersome and impractical and thus the major chal-lenge of such work is a) finding a simplified, useful ap-proximate expression by invoking various assumptionsb) determining the exact conditions for which the asso-ciated assumptions are valid. These two goals and thedescribed basic methodology guide the work which fol-lows throughout this paper. In general, we do not pro-vide detailed derivations in the main text for the sake ofbrevity, but all relevant derivations are included in detailin the appendices. We begin our exploration of various AP effects by consid-ering that the idealized assumption M transiter (cid:28) M (cid:63) isinvalid (the masses of transiting object and star respec-tively). We note that Dawson & Johnson (2012) brieflycommented on this possibility previously (see § M transiter term in the derivationof the stellar density returns the result ρ (cid:63), obs = ρ (cid:63), true + p ρ transiter , (5)where ρ transiter is the mean density of the transitingobject (usually this is a transiting planet but the ex-pressions are valid for eclipsing binaries too). This resultimplies that (cid:32) ρ (cid:63), obs ρ (cid:63), true (cid:33) = 1 + p ρ transiter ρ (cid:63), true , (6) which we may re-express as (cid:32) ρ (cid:63), obs ρ (cid:63), true (cid:33) PM = 1 + M transiter M (cid:63) , (7)where we use the subscript “PM” as an acronymfor the photo-mass effect. Negating the planetary masstherefore causes us to overestimate ( ρ (cid:63), obs /ρ (cid:63), true ). For aconfirmed/validated exoplanet, this effect will be (cid:46) One of the most critical assumptions in the derivationof ρ (cid:63), obs is that the brightness variations observed aredue to the host star alone, which means that the staris unblended. Blend sources come in many varieties in-volving triple and binary stellar configurations (Torreset al. 2011; Hartman et al. 2011) as well as even self-blending due to a hot compact object such as a white-dwarf or even a hot-Jupiter (Kipping & Tinetti 2010).Blend sources are the astrophysical bottleneck in con-firming/validating the thousands of planetary candidatesfound by the Kepler Mission (Morton & Johnson 2011;Fressin et al. 2011).We define the blend factor, B , in this work as theratio of the total flux to that of the target’s flux, via B ≡ F (cid:63) + F blend F (cid:63) , (8)where F (cid:63) is the flux received from the target and F blend is the sum of all extra contaminating components.In Appendix A, we show that if we assume ( a/R (cid:63) ) (cid:29) (1 + p ) , the effect of a blend may be expressed as (seeEquation A18): (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) PB = B − / (cid:32) (1 + √B p obs ) − b (1 + p obs ) − b (cid:33) / . (9)Equation 9 is maximized for b obs → (1 − p obs ) and p true → B (cid:39) a/R (cid:63) ) (cid:29) (1 + p ) , may also be expressed as (cid:16) P days (cid:17) / (cid:29) . (cid:16) ρ (cid:63), true g cm (cid:17) − / . (10)It can be seen from the above that this conditionshould be satisfied for all but the very shortest of or-bital periods (e.g. Kepler-78b; Sanchis-Ojeda et al. 2013).Since all blend sources must satisfy B > c (cid:13) , 1–26 sterodensity Profiling measure the blend factor B by inverting Equation 9. Asdiscussed in detail in Appendix A5, inverting Equation 9yields a quadratic equation with two valid roots: B + , − = 14 p (cid:32) − p obs + (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / [(1 + p obs ) − b ] ± (cid:34)(cid:16) p obs (cid:104) (2 + p obs ) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / − (cid:105) + (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / (1 − b ) (cid:17) − p (1 − b ) (cid:35) / (cid:33) . (11)The B + , − functions are plotted in Figure 1 for dif-ferent input parameters. There are several key observa-tions of the expression. Firstly, for a known p obs and b obs ,( ρ (cid:63), obs /ρ (cid:63), true ) is always bound by the range: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) min (cid:54) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) max , (12)where (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) min = (cid:32) p obs (1 + (cid:112) − b )(1 + p obs ) − b (cid:33) / , (13) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) max = 1 . (14)Curiously then, there is both an upper and lowerlimit on the range of ρ (cid:63), obs values a blend can produceany observation outside of this range cannot be due tothe photo-blend effect only.The second important observation is although thesolution for B is bi-modal, it is actually uni-modal formost ( ρ (cid:63), obs /ρ (cid:63), true ) inputs. Specifically, as shown in Ap-pendix A5, the B + is unphysical most inputs. This is alsoillustrated in Figure 1 by the gray dotted line. In practicethen, only a small range of parameter space is bi-modal,which occurs when: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) min (cid:54) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) B + , max , (15)where (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) B + , max = (cid:32) (4 − b ) p obs (1 + p obs ) − b (cid:33) / . (16)In fact, as visible in Figure 1, this bi-modal rangehas zero volume as b obs → (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) min = (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) B + , crit in this limit. In summary then, we have: B = no roots if 0 < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) min B − or B + if (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) min < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) B + , crit B − if (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) B + , crit < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) < < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) < ∞ (17) Starspots, networks and plages are thought to form bystellar magnetic fields generated by cyclonic turbulencein the outer convection zone of cool stars penetratingthe stellar atmosphere (Berdyugina 2005). Starspots arethought to be a particularly common outcome of this pro-cess and continuous photometric monitoring reveals theirsignature as rotational modulations, which has allowedfor the determination of rotation periods for thousandsof stars (Basri et al. 2011; Walkowicz & Basri 2013).Whilst large spots which are occulted by the tran-siting object are easy to identify and remove, unoccultedspots are more challenging and perturb the transit depthas pointed out by Czesla et al. (2009). We define theact of unocculted starspots perturbing the observed tran-sit depth, and thus the observed stellar density, as the“photo-spot” effect.Equation 9 reveals that since B (cid:62) B < ρ (cid:63), obs /ρ (cid:63), true ) >
1. Kipping (2012) showed thatthe effect of the transit depth is given by δ obs δ true = F (cid:63) (unspotted) F (cid:63) (spotted) ,δ obs δ true = 11 − A spots , (18)where F (cid:63) is the flux from the star and the unspot-ted case corresponds to the flux an observer would see ifone took the actual starspot population and shrunk theirsizes to zero. The second line re-writes this expression bydefining A spots as the effective normalized photometricamplitude of the rotational modulations. For a rotatingstar with one to a few major spots, there will be timeswhen all of the spots in view and times when no spotsare present, giving rise to quasi-periodic transit depthvariations (T δ V). We assume such a rotation period a)much longer than the transit duration, b) much shorterthan the baseline of observations and c) has no commen-surability with the transiting body’s orbital period. If wetreat F (cid:63) (spotted) as behaving like a Fourier series of har-monic components, then the average effect on the transitdepths (i.e. the folded transit light curve depth) wouldbe ¯ δ obs δ true (cid:39) − A spots / , (19)The photo-spot effect is illustrated in Figure 2,where the T δ Vs give rise an apparently increased depthin the folded light curve. The depth ratio, (¯ δ obs /δ true ), isequivalent to B − using our definition of the blend factorin Equation 8. Exploiting this trick, one may write thata spot behaves like a blend with a blend factor, B spot ,given by c (cid:13)000
1. Kipping (2012) showed thatthe effect of the transit depth is given by δ obs δ true = F (cid:63) (unspotted) F (cid:63) (spotted) ,δ obs δ true = 11 − A spots , (18)where F (cid:63) is the flux from the star and the unspot-ted case corresponds to the flux an observer would see ifone took the actual starspot population and shrunk theirsizes to zero. The second line re-writes this expression bydefining A spots as the effective normalized photometricamplitude of the rotational modulations. For a rotatingstar with one to a few major spots, there will be timeswhen all of the spots in view and times when no spotsare present, giving rise to quasi-periodic transit depthvariations (T δ V). We assume such a rotation period a)much longer than the transit duration, b) much shorterthan the baseline of observations and c) has no commen-surability with the transiting body’s orbital period. If wetreat F (cid:63) (spotted) as behaving like a Fourier series of har-monic components, then the average effect on the transitdepths (i.e. the folded transit light curve depth) wouldbe ¯ δ obs δ true (cid:39) − A spots / , (19)The photo-spot effect is illustrated in Figure 2,where the T δ Vs give rise an apparently increased depthin the folded light curve. The depth ratio, (¯ δ obs /δ true ), isequivalent to B − using our definition of the blend factorin Equation 8. Exploiting this trick, one may write thata spot behaves like a blend with a blend factor, B spot ,given by c (cid:13)000 , 1–26 David M. Kipping
Figure 1.
The Photo-blend Effect:
Blends, or uncorrected contaminated light, always cause one to underestimate the stellardensity, plotted here on the x -axis as ( ρ (cid:63), obs /ρ (cid:63), true ) . One may solve for the blend factor, B , to aid in validating candidateplanets, yielding two analytic roots shown by the curved black ( B + ) and black-dashed ( B − ) lines for a range of apparent impactparameters, b obs , and ratio-of-radii, p obs . The B + root is only physically valid between the point of inflection of the contours(traced by the black dot-dashed line) and the dotted gray line. B spot (cid:39) − A spots , B spot (cid:39) (cid:32) F (cid:63) (spotted) F (cid:63) (unspotted) (cid:33) . (20)Equipped with Equation 20, one may now computethe consequences on the stellar density using the sameexpressions derived earlier for the photo-blend effect in § (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) PS = lim B→B spot (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) PB (21)We note that plotting the B + root for( ρ (cid:63), obs /ρ (cid:63), true ) > B > B − root for the photo-spot effect. As withthe photo-blend effect, the same conditions apply forthe application of these analytic photo-spot equations: c (cid:13) , 1–26 sterodensity Profiling (cid:16) P days (cid:17) / (cid:29) . (cid:16) ρ (cid:63), true g cm (cid:17) − / . (22)Typically, even a heavily spotted star will be inthe range A spots (cid:46)
20% and usually (cid:46) ρ (cid:63), obs /ρ (cid:63), true ) at the same order-of-magnitude level as the normalized rotational mod-ulations amplitude. The maximum AP deviation viathis effect can be evaluated by computing the limit for b obs → (1 − p obs ) and p true →
1. For an extreme 20% spotamplitude we obtain an AP effect of order O [10 − ], andfor a typical 1% spot amplitude this becomes O [10 − ]. Inprinciple, it is possible to correct for the photo-spot effectusing rotational modulation data, although this can bechallenging (Kipping 2012) and such effort should be putin the context of the expected magnitude of this effect. Transit timing variations (TTVs) have been revealed bythe
Kepler Mission to be a fairly common occurrence inplanetary systems (Ford et al. 2012; Mazeh et al. 2013)with ∼
10% showing significant TTVs. TTVs of low am-plitude can be difficult to infer by fitting individual tran-sits and yet if we ignore their presence they will system-atically bias the derived transit parameters. An objectwith low-amplitude TTVs ( A TTV < T ) with N (cid:29) T and T dura-tions. Naturally, this will feed into the derived impactparameter, scaled semi-major axis and light curve de-rived stellar density.In Appendix D, we derive the full consequences ofunaccounted TTVs on the derived transit parameters.The effect on ρ (cid:63) depends upon the true impact parame-ter but unfortunately the impact parameter is also cor-rupted by the TTVs. One way round this is to considerthe worst-case scenario where ρ (cid:63), obs is most discrepantfrom ρ (cid:63), true , which occurs for b = 0. In this case, onefinds a simple form for the photo-timing effect: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) PT (cid:62) (cid:32) pp + nA TTV ( a/R (cid:63) ) (cid:33) / , (23)where n = 2 π/P , 2 A TTV is the peak-to-peak TTVamplitude, ( a/R (cid:63) ) implicitly refers to ( a/R (cid:63) ) true and weuse the (cid:62) symbol since the calculation is computed forthe extreme case of b = 0. In Appendix D, we show thatthis expression is a valid approximation for:( a/R (cid:63) ) (cid:29) , (24)2 A TTV (cid:28) T . (25)The first condition may also be re-expressed in phys-ical units as (cid:16) P days (cid:17) / (cid:29) . (cid:16) ρ (cid:63), true g cm (cid:17) − / . (26)In general, one expects TTVs to be detectable bycareful inspection of the data. In the case that significantTTVs are present, an accurate light curve derived stellardensity could be derived by either using a model whichallows for unique transit times or using a physical modelwhich accounts for TTVs (i.e. a photodynamical model),provided the physical model well-explains the data.If no significant TTVs are detected, or an observeropts to try and remove the best-fitting TTVs and thenre-fit assuming a linear ephemeris, the light curve de-rived stellar density can still be affected by unseen low-amplitude TTVs. In principle, one expects to be able toexclude TTVs up to some maximum amplitude level to1-, 2-, 3- (etc) σ confidence. In essence, this means thatthe uncertainty on ρ (cid:63), obs will be underestimated. How-ever, using our expressions, it is possible to quantify thisunaccounted-for uncertainty in the extreme case occur-ring for b true = 0: σ ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:46) . G / ρ / (cid:63), obs pP / σ τ N / , (27)where σ τ is the typical timing uncertainty on eachtransit and N is the number of transits observed. As anexample, consider a planet with P = 10 days, p = 0 . σ max( ρ (cid:63), obs /ρ (cid:63), true ) (cid:46) ρ (cid:63), obs . This demonstratesthe photo-timing effect leads to inflated errors on thestellar density and caution must be taken in interpret-ting small discrepancies. For much larger timing errorsof σ τ (cid:39)
10 minutes, the effect can be estimated to be O [10 ]. Transit durations variations (TDVs) are another exam-ple of a dynamical effect which will alter the shape of afolded transit light curve, if left unaccounted for. TDVswere first posited to be a signature of exomoons (Kip-ping 2009a,b) but have since been demonstrated to bealso possible in strongly interacting multi-planet sys-tems too (Nesvorn´y et al. 2012). TDVs come in two fla-vors, velocity-induced transit duration variations, TDV-V, and transit impact parameter induced transit du-ration variations, TDV-TIP (Kipping 2009a,b). In thiswork, we focus on the more dominant component ofTDV-V.TDV-Vs essentially stretch and squash the widthof the transit shape and a well sampled periodic set oflight curves with TDV-Vs will exhibit a deformed foldedtransit shape, if neglected. This is illustrated in Figure 4, c (cid:13)000
10 minutes, the effect can be estimated to be O [10 ]. Transit durations variations (TDVs) are another exam-ple of a dynamical effect which will alter the shape of afolded transit light curve, if left unaccounted for. TDVswere first posited to be a signature of exomoons (Kip-ping 2009a,b) but have since been demonstrated to bealso possible in strongly interacting multi-planet sys-tems too (Nesvorn´y et al. 2012). TDVs come in two fla-vors, velocity-induced transit duration variations, TDV-V, and transit impact parameter induced transit du-ration variations, TDV-TIP (Kipping 2009a,b). In thiswork, we focus on the more dominant component ofTDV-V.TDV-Vs essentially stretch and squash the widthof the transit shape and a well sampled periodic set oflight curves with TDV-Vs will exhibit a deformed foldedtransit shape, if neglected. This is illustrated in Figure 4, c (cid:13)000 , 1–26 David M. Kipping
Figure 2.
The Photo-spot Effect:
By neglecting to correct for transit depth variations (T δ V) due to unocculted spots, a foldedtransit light curve will exhibit deformation leading to the erroneous retrieval of the basic transit parameters, including the observedstellar density, ρ (cid:63), obs . Here, the black line represents the true original signal, the gray lines are 100 examples of the signal withunaccounted for sinusoidal T δ Vs and the black-dashed line is the naively folded transit light curve, exhibiting sizable deformation. where the composite light curve is deformed in a simi-lar way to that caused by periodic TTVs earlier in Fig-ure 3. We consider the TDVs to be due to the velocityof a planet varying periodically between the extrema of v min = v (1 − A TDV ) and v max = v (1 + A TDV ), wherethe “0” subscript denotes the parameter’s value in theabsence of TDVs. Since the durations are inversely pro-portional to the velocity of the planet, then the durationsvary over time t over the range: T , (1 − A TDV ) (cid:54) T ( t ) (cid:54) T , (1 + A TDV ) . (28)The A TDV term therefore defines the relativechanges in the duration, and not the absolute changes,which is the more natural expression of TDV-Vs. Usingthis model, we derive the effect of periodic TDVs on thelight curve derived stellar density in Appendix E to be: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) PD = (cid:32) ( a/R (cid:63) ) p + 4 A b p + 2 A TDV [(1 − p ) − b (1 + p )]( a/R (cid:63) ) [ p + 4 A p + 2 A TDV (1 + p − b )] (cid:33) / , (29) where ( a/R (cid:63) ) is ( a/R (cid:63) ) true and can be estimatedas [( GP ρ (cid:63) ) / (3 π )] / . In Appendix E, we show that theabove is valid when ( a/R (cid:63) ) (cid:29) , (30) A TDV (cid:28) . (31)As with the photo-timing effect, the photo-durationeffect can be thought of as imparting an error term on theobserved stellar density. We are unable to find a simpleform for the resulting expression though and so suggestobservers use: σ PD( ρ (cid:63), obs /ρ (cid:63), true ) = 1 − (cid:34) lim A TDV → σ A TDV (cid:32) ρ (cid:63), obs ρ (cid:63), true (cid:33) PD (cid:35) . (32)As with the photo-timing effect error, we demon-strate the above by considering the same example of aplanet with P = 10 days, p = 0 . b = 0 ( T , = 4 . σ max( ρ (cid:63), obs /ρ (cid:63), true ) = 10 . c (cid:13) , 1–26 sterodensity Profiling Figure 3.
The Photo-timing Effect:
If even low-amplitude transit timing variations (TTV) are negated, a folded transitlight curve will exhibit deformation leading to the erroneous retrieval of the basic transit parameters, including the observedstellar density, ρ (cid:63), obs . Here, the black line represents the true original signal, the gray lines are 100 examples of the signal withunaccounted for sinusoidal TTVs and the black-dashed line is the naively folded transit light curve, exhibiting sizable deformation. Therefore, in this example, the photo-duration effect im-parts approximately the same level of uncertainty intothe observed stellar density as the photo-timing effectdoes. However, unlike TTVs, TDVs are considerablyrarer in the database of known exoplanets with only afew examples and so the a-priori probability of hiddenTDVs is clearly distinct to that from timing variations.
The effect of eccentricity is the most well-studied aster-odensity profiling effect. Dawson & Johnson (2012) re-fer to this asterodensity profiling effect as the “photo-eccentric effect”, as we do so here. The first explicitderivation of the effect of eccentricity is given in Kip-ping et al. (2012) who find (cid:16) ρ obs ρ true (cid:17) PE = Ψ , (33)where Ψ ≡ (1 + e sin ω ) (1 − e ) / . (34)Despite the expression already existing in the litera-ture, we are unaware of any investigations regarding therange of parameters which the approximation shown inEquation 33 is valid. In Appendix B, we present a de-tailed investigation of this and surmise that the above isvalid for: ( a/R (cid:63) ) (cid:29) (cid:32) e − e (cid:33) , (35)which may also be expressed in physical units as (cid:32) P days (cid:33) / (cid:29) . (cid:32) ρ (cid:63), true g cm − (cid:33) − / (cid:32) e − e (cid:33) . (36)As pointed out in numerous previous works (Burke2008; Kipping 2008; Winn 2010; Dawson & Johnson2012), if a planet on an eccentric orbit is observed totransit, then a-priori it is more probable that 0 < ω (cid:54) π than π < ω (cid:54) π . This is because the geometric transitprobability is given by c (cid:13)000
The effect of eccentricity is the most well-studied aster-odensity profiling effect. Dawson & Johnson (2012) re-fer to this asterodensity profiling effect as the “photo-eccentric effect”, as we do so here. The first explicitderivation of the effect of eccentricity is given in Kip-ping et al. (2012) who find (cid:16) ρ obs ρ true (cid:17) PE = Ψ , (33)where Ψ ≡ (1 + e sin ω ) (1 − e ) / . (34)Despite the expression already existing in the litera-ture, we are unaware of any investigations regarding therange of parameters which the approximation shown inEquation 33 is valid. In Appendix B, we present a de-tailed investigation of this and surmise that the above isvalid for: ( a/R (cid:63) ) (cid:29) (cid:32) e − e (cid:33) , (35)which may also be expressed in physical units as (cid:32) P days (cid:33) / (cid:29) . (cid:32) ρ (cid:63), true g cm − (cid:33) − / (cid:32) e − e (cid:33) . (36)As pointed out in numerous previous works (Burke2008; Kipping 2008; Winn 2010; Dawson & Johnson2012), if a planet on an eccentric orbit is observed totransit, then a-priori it is more probable that 0 < ω (cid:54) π than π < ω (cid:54) π . This is because the geometric transitprobability is given by c (cid:13)000 , 1–26 David M. Kipping
Figure 4.
The Photo-duration Effect:
If even low-amplitude transit duration variations (TDVs) are negated, a folded transitlight curve will exhibit deformation leading to the erroneous retrieval of the basic transit parameters, including the observedstellar density, ρ (cid:63), obs . Here, the black line represents the true original signal, the gray lines are 100 examples of the signal withunaccounted for sinusoidal TDVs and the black-dashed line is the naively folded transit light curve, exhibiting sizable deformation. P( b (cid:54)
1) = 1( a/R (cid:63) ) 1 + e sin ω − e (37)and so P(0 < ω (cid:54) π | b (cid:54) π < ω (cid:54) π | b (cid:54)
1) = π + 2 eπ − e (38)which is greater than 1 for all 0 < e (cid:54)
1. Note thatthe exact ratio cannot be estimated without assumingsome prior distribution for the eccentricity. The conse-quence of this is that Ψ > < tend to overes-timate ( ρ (cid:63), obs /ρ (cid:63), true ) whereas all of the previous effectsdiscussed, except the photo-mass effect, underestimatethe density. However, it is also worth noting that even formoderately high eccentricities of say e ∼ .
5, the odds-ratio quoted above is ∼ The photo-eccentric effect directly reveals Ψ, which is afunction of both e and ω . Ideally, one wishes to obtaininformation on both e and ω in isolation, but purely froman information theory perspective it is obvious this idealcan never be truly realized, since we have one measure-ment and two unknowns. Progress can be made by con-sidering the minimum eccentricity. In the Appendix C,we show that the minimum eccentricity can be derivedin the case of SAP and provide a single-domain functionfor e min as e min = (cid:32) (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / + 1 (cid:33) H (cid:104)(cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) − (cid:105) + (cid:32) (1 − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / )(1 − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / + (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / )1 + (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) (cid:33) H (cid:104) − (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17)(cid:105) , (39)where H [ x ] is the Heaviside Theta function. We notethat previous authors have derived or discuss double-domain functions for e min such as Barnes (2007) and c (cid:13) , 1–26 sterodensity Profiling Kane et al. (2012). The single-domain function presentedhere simply combines the two domains using HeavisideTheta functions and uses stellar density as the observ-able rather than durations. We also stress that e min ispurely a function of ( ρ (cid:63), obs /ρ (cid:63), true ) and no other terms.It is therefore possible to analytically calculate the un-certainty on e min using quadrature: σ e min = 43 (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) − / (cid:32) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / (cid:33) − σ ρ (cid:63), obs /ρ (cid:63), true , (40)where σ e min and σ ρ (cid:63), obs /ρ (cid:63), true are the uncertaintieson the minimum eccentricity and ratio of the observedto true stellar density respectively.The simple e min function is visualized in Figure 5,where one can see the PE effect can induce AP deviationsup to O [10 ]. In this figure, we also over-plot Kepler Ob-jects of Interest (KOIs) with asteroseismologically mea-sured ρ (cid:63), true values from Huber et al. (2013). The ρ (cid:63), obs term is computed for each KOI using the MAST archivaldatabase entries of ( a/R (cid:63) ) and P . In principle, objectson the left-hand side (LHS) may be blends (since onedoes not expect a high proportion of eccentric planets onthis side) and objects on the right are genuinely eccentricKOIs (therefore assuming that the photo-timing, photo-duration, photo-spot and photo-mass effects are minor).However, we caution that the large number of multison the LHS, suggesting false-positive blended systems, ishighly inconsistent with the expected low false-positiverates of multi-planet systems (Lissauer et al. 2012). Wetherefore advocate independent checks of these ρ (cid:63), obs val-ues before drawing any conclusions, which is outside thescope of this work. Calculating the minimum eccentricity using Equation 39is distinct from the strategy on the photo-eccentric effectby Dawson & Johnson (2012) and Dawson et al. (2012)who propose marginalizing over ω , much like a nuisanceparameter. The major advantage over marginalizing over ω is that one naturally incorporates the geometric transitprobability effect and derives a singular estimate for e .This is useful since e represents the most physically usefulparameter with respect to formation/evolution models(Ford & Rasio 2008; Juri´c & Tremaine 2008; Socrates etal. 2012; Dong et al. 2013).However, there are several drawbacks of this ap-proach compared to simply computing e min using Equa-tion 39. Firstly, one can only achieve this feat by as-suming an a-priori distribution for the eccentricity sincethe geometric transit probability is functionally depen-dent on both e and ω , as discussed earlier. Therefore,the derived e value is fundamentally dependent upon the http://archive.stsci.edu/kepler/koi/search.php assumed prior distribution for e , which is somewhat cir-cular logic. In practice, Dawson & Johnson (2012) foundvarying the priors on e imposes only small changes inthe derived posterior distributions of e , yet this we pre-dict that this is only likely true where the data over-whelms the priors such as the cases considered by Daw-son & Johnson (2012) of highly eccentric systems causing( ρ (cid:63), obs /ρ (cid:63), true ) (cid:29) e space is over e min < e <
1, since e < e , this will cause the marginalized e value to lie some-where inbetween these two extrema and lead to elevatederror bars relative to e min to accommodate this marginal-ization. Systems with a high e min will therefore appearto provide relatively small errors on the marginalized e ,purely because there is “less room” between e min andunity. An example of this is evident with the high e min system of HD 17156b reported in Dawson & Johnson(2012) ( e min ∼ .
6) giving e = 0 . +0 . − . whereas the e min ∼ e = 0 . +0 . − . (Kipping et al. 2013), de-spite both being bright targets with asteroseismologyand high-quality photometry. In general then, we advo-cate at least providing the community with both the con-strained, prior-independent e min term in addition to thelossy, marginalized e .Thirdly, the simple analytic form of our expressionfor e min makes it attractive for rapid calculation on hun-dreds/thousands of systems. Transits may be fitted en-masse assuming a circular orbit and then e min is easilycomputed without any tacit assumption on the e dis-tribution. An alternative but equivalent approach wouldbe to compile a database of T , obs , T , obs and δ obs fromwhich one can also proceed to compute e min . Addition-ally, provided one knows the uncertainties on ρ (cid:63), obs and ρ (cid:63), true , then the uncertainty on e min is easily recovered ina single expression given by Equation 40, at least underthe assumption of no other AP effects. We propose thatthis would be an advantageous strategy for upcomingtransit survey missions, such as TESS. Let us define a “false-positive” planetary candidate to beone which orbits a different star to that for which we havean independent measure of the stellar density. In sucha case, then it should be obvious that the two densityestimates need not agree and can be grossly different (seeSliski & Kipping 2014 for examples of such cases). Theexact difference will depend upon the spectral types ofthe two stars and the flux ratios. In this scenario, we haveno information on ρ (cid:63), true , since the independent measurecorresponds to a different star. We also know that thestar hosting the transiting body must be heavily blended c (cid:13)000
6) giving e = 0 . +0 . − . whereas the e min ∼ e = 0 . +0 . − . (Kipping et al. 2013), de-spite both being bright targets with asteroseismologyand high-quality photometry. In general then, we advo-cate at least providing the community with both the con-strained, prior-independent e min term in addition to thelossy, marginalized e .Thirdly, the simple analytic form of our expressionfor e min makes it attractive for rapid calculation on hun-dreds/thousands of systems. Transits may be fitted en-masse assuming a circular orbit and then e min is easilycomputed without any tacit assumption on the e dis-tribution. An alternative but equivalent approach wouldbe to compile a database of T , obs , T , obs and δ obs fromwhich one can also proceed to compute e min . Addition-ally, provided one knows the uncertainties on ρ (cid:63), obs and ρ (cid:63), true , then the uncertainty on e min is easily recovered ina single expression given by Equation 40, at least underthe assumption of no other AP effects. We propose thatthis would be an advantageous strategy for upcomingtransit survey missions, such as TESS. Let us define a “false-positive” planetary candidate to beone which orbits a different star to that for which we havean independent measure of the stellar density. In sucha case, then it should be obvious that the two densityestimates need not agree and can be grossly different (seeSliski & Kipping 2014 for examples of such cases). Theexact difference will depend upon the spectral types ofthe two stars and the flux ratios. In this scenario, we haveno information on ρ (cid:63), true , since the independent measurecorresponds to a different star. We also know that thestar hosting the transiting body must be heavily blended c (cid:13)000 , 1–26 David M. Kipping
Figure 5.
The Photo-eccentric Effect:
The minimum orbital eccentricity function, defined in Equation 39, plotted with respectto its only dependent variable, ( ρ (cid:63), obs , ρ (cid:63), true ) . The arrows correspond to real KOIs with known asteroseismology measurementsavailable, where blue are singles and red are multis. We also mark the directions in which the other asterodensity profiling effectsact. and so the photo-blend effect must be acting. If we ignorethe other AP effects, we may recall from Equation 14 thata limit exists on the maximum AP deviation due to thePB effect: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) (cid:62) (cid:32) p obs (1 + (cid:112) − b )(1 + p obs ) − b (cid:33) / . (41)This may be re-expressed to constrain the unknownquantity ρ (cid:63), true via: ρ (cid:63), alt (cid:62) ρ (cid:63), obs (cid:32) (1 + p obs ) − b p obs (1 + (cid:112) − b ) (cid:33) / , (42)where we replace ρ (cid:63), true with ρ (cid:63), alt to stress that thetransiting body is orbiting an alternative star. We donot refer to this scenario as a “photo- name ” effect, sinceunlike the other cases no independent information on thetrue stellar density is available. However, Equation 42,which is only valid in the absence of the PE, PT, PD,PM and PS effects, may be of use to observers vettingplanetary candidates. In this work, we have shown that at least five other aster-odensity profiling effects exist in addition to the photo-eccentric effect, which are summarized in the “cheatsheet” of Figure 6. Since a number of phenomena caninduce significant changes to the light curve derived stel-lar density, great care must be taken not to over-interpretany deviations as implying high eccentricity where noneexists. Equivalently, one must be wary of interpreting alack of significant discrepancies as implying most planetsorbit on near-circular orbits. Put succinctly, the eccen-tricity distribution can only be extracted using AP witha careful consideration of the prior distributions for theother AP effects (e.g. the photo-blend effect, the photo-timing effect, etc). We would argue that any eccentricitydistributions purported without such due diligence can-not be considered physically representative of the truesample. This statement is justified by the fact that notonly are the other AP effects significant at the typicalmeasurement uncertainties, but they will impart system- c (cid:13) , 1–26 sterodensity Profiling atic shifts to any naively computed e distribution, sincethe parameters upon which they depend are often skewedin one direction (e.g. for the photo-blend effect the blendparameter must always be > (cid:32) P days (cid:33) / (cid:29) . (cid:32) ρ (cid:63), true g cm − (cid:33) − / (cid:32) e − e (cid:33) . (43)The above has a very steep dependency on e andrapidly rises as e approaches unity, due to the (1 − e ) − term. This is particularly salient in light of the pre-diction and subsequent observational search for proto-hot Jupiters on super-eccentric orbits by Socrates et al.(2012) and Dawson et al. (2013) respectively. For exam-ple, if we wish to exploit the analytic photo-eccentriceffect to search for objects with e = 0 . P (cid:29)
114 days i.e. we need P (cid:38) ρ (cid:63), obs /ρ (cid:63), true ) dis-crepancy, but one cannot reliably use the photo-eccentricequations to back out e or e min .For individual systems, priors on these other APeffects are likely less relevant since for many well-characterized transiting planet systems there are of-ten additional observational constraints on many of theterms which affect the various AP effects e.g. rotationalmodulations, transit timing variations, adaptive optics,centroid offsets, etc. Therefore, SAP still presents ar-guably the most feasible technique for measuring the ec-centricity of small, habitable-zone planets with currenttechniques (e.g. see the recent demonstration with thehabitable-zone planet Kepler-22b, Kipping et al. 2013). Throughout this work we have derived analytic resultsfor various AP effects in the Single-body Asteroden-sity Profiling (SAP) paradigm, since all results comparethe observed stellar density to some independent “true”measure. However, it is trivial to extend our results tothe case of Multi-body Asterodensity Profiling (MAP),which was first discussed in Kipping et al. (2012). Bycomparing the observed stellar density between transit-ing planets j and k , and assuming the objects orbit thesame star, one is able to extract information on the stateof a system. Since the issue of eccentricity is discussed indetail in Kipping et al. (2012), we do not repeat the ar-guments made in that work but briefly summarize thatthe authors found an analytic minimum constraint onthe pair-wise sum of eccentricities for planets j and k iseasily derived using MAP: e j + e k (cid:62) Θ jk − , (44) where Θ jk ≡ (cid:32) ρ (cid:63), obs ,j ρ (cid:63), obs ,k (cid:33) / . (45)The two-thirds index was chosen since it naturallyremoves a three-halves index in the expression for thephoto-eccentric effect. As shown in this work, it can beseen that the photo-blend effect also happens to be de-scribed by a three-halves power and thus the same Θ jk definition is valuable in performing MAP for blend anal-ysis. This is pertinent since the photo-blend effect caninduce very large deviations in the observed stellar den-sities and, for heavily blended systems, this effect dom-inates AP. Using Equation 9 then, one may write theMAP blend equation as:Θ jk = [(1 + √B p obs ,j ) − b ,j ][(1 + p obs ,k ) − b ,k ][(1 + √B p obs ,k ) − b ,k ][(1 + p obs ,j ) − b ,j ] . (46)It is possible to invert this equation and actuallysolving for √B , yielding two roots from a quadratic equa-tion: (cid:113) B MAP ± = p obs ,j α obs ,k − Θ jk p obs ,k α obs ,j ± (cid:112) D jk Θ jk p ,k α obs ,j − p ,j α obs ,k , (47)where we make the substitutions α k = (1 + p obs ,k ) − b ,k , (48) α j = (1 + p obs ,j ) − b ,j , (49) D jk = b ,k p ,k α ,j Θ jk − b ,j p ,j α ,k − α obs ,k α obs ,j Θ jk (cid:104) p obs ,k p obs ,j − p ,k (1 − b ,j ) − p ,j (1 − b ,k ) (cid:105) . (50)As useful check of these expressions is to evaluatethem in the limit of Θ jk →
1, which would be the ob-served value in the absence of any blending:lim Θ jk → (cid:113) B MAP − = 1 , (51)lim Θ jk → (cid:113) B MAP+ = (cid:104) p obs ,k (2 + p obs ,k )(1 − b ,j ) − p obs ,j (2 + p obs )(1 − b ,k ) (cid:105)(cid:104) p ,j (1 − b ,k ) − p ,k (1 − b ,j ) + 2 p obs ,k p obs ,j ( p obs ,j − p obs ,k ) (cid:105) − . (52)Therefore, as expected, we recover the B = 1 so-lution corresponding to no blending. However, the sec-ond root cannot be trivially dismissed and is physi-cally plausible. Generating uniform random values for0 < p obs ,j <
1, 0 < p obs ,k <
1, 0 < b obs ,j < (1 − p obs ,j ) c (cid:13)000
1, 0 < b obs ,j < (1 − p obs ,j ) c (cid:13)000 , 1–26 David M. Kipping
Figure 6.
Asterodensity Profiling “Cheat Sheet”:
Summary of the analytic formulae for the various AP effects derived inthis work, including the supported parameter range for their applicability. Red boxes provide approximate order-of-magnitude foreach effect. All effects also assume < b < (1 − p ) i.e. a “flat-bottomed” transit. c (cid:13) , 1–26 sterodensity Profiling and 0 < b obs ,k < (1 − p obs ,k ), we find that 68.3% ofthe samples lie in the range 1 . < B MAP+ < . jk →
1, the B MAP+ solution does not produce grossly large blend fac-tors which can be easily dismissed as unphysical.We therefore conclude that for 2-planet systems,blend analyses with MAP will be challenged by this ap-parent bi-modality. However, with n (cid:62) n ! / (2!( n − jk and the true solution for B will be recovered in one of thetwo roots every time. Therefore, it should be possible toidentify which root corresponds to the true solution in 3or more planet solutions by root comparison. We leavemore detailed investigations of MAP blend analysis tofuture studies. We hope that the investigations presented in this paperprovide the foundational analytic theory for asteroden-sity profiling, but we are acutely aware that there is agreat deal of theoretical and observational work still toaccomplish in this new area of study. To begin with, thereare numerous ignored effects known to distort the transitlight curve and thus have the potential to impart AP sig-natures, such as planetary rings (Ohta et al. 2009; Barnes& Fortney 2004), planetary oblateness (Carter & Winn2010), atmospheric lensing (Hui & Seager 2002), etc. Wealso did not consider cases where p >
1, such as totaleclipses of white dwarfs discussed in Agol (2011).In order to retrieve the eccentricity distribution us-ing AP, we suggest that significant work is needed tounderstand the blend distribution, TTV distribution,starspot distribution, etc in order to adequately de-convolve the contribution from other AP effects. Well-characterized individual systems will likely be less de-pendent upon these prior distributions and so immedi-ate observational progress can surely be made here (e.g.Dawson et al. 2012; Kipping et al. 2013). In such cases,we would advocate research into how well the blend fac-tor can be constrained and whether systems can be prac-tically validated using AP.Despite these challenges, we envisage that AP canbe a powerful tool for archival
Kepler data and for theforthcoming TESS mission, for both measuring the ec-centricity distribution and validating/vetting planetarycandidates. This work also underscores the valuable sym-biosis between exoplanet transits and asteroseismologyfor characterizing distant worlds.
ACKNOWLEDGMENTS
DMK would like to thank D. Sliski, S. Ballard and J.Irwin for very helpful conversations in the preparation ofthis work. Thanks to the anonymous reviewer for theirconstructive comments. DMK is supported by the NASASagan Fellowship.
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Transits and Occultations , EXO-PLANETS, University of Arizona Press; ed: S. Seager c (cid:13) , 1–26 sterodensity Profiling APPENDIX A: DERIVATION OF THEPHOTO-BLEND EFFECTA1 Ratio-of-radii bias
We here provide a formal derivation for the photo-blendeffect, followed by in subsequent appendices by other rel-evant derivations of important results presented in thispaper. All derivations, unless otherwise stated, will fol-low the methodology outlined in § B . It therefore follows that the derived ratio-of-radii, p , is affected via p obs = p true / √B , (A1)where we use the subscripts “true” and “obs” todistinguish between the truth and that which one naivelyadopts the standard simple assumptions of no blend, acircular orbit, etc. The true value may be therefore beretrieved using p true = p obs √B . (A2) A2 Impact parameter bias
As stated earlier, we will ignore all other AP effects inwhat follows. Accordingly, one may define the transitimpact parameter as a function of just three terms T , T and p , as demonstrated by Seager & Mall´en-Ornelas(2003). In this framework, T is the first-to-fourth con-tact duration and T is the second-to-third contact du-ration. Critically, these durations are unaffected by theact of a blend (Kipping & Tinetti 2010). The same state-ment can also be said of the the orbital period which iscalculated by the interval between transits: T , obs = T , true = T ,T , obs = T , true = T ,P obs = P true = P, (A3)where we drop the explicit “true” subscript on theright-hand side (RHS). We follow this pattern in whatfollows, where the reader should interpret any term miss-ing an explicit true/obs subscript to imply that we arereferring to the true value. Having now defined the effectof blends on each of the key observable terms, we maynow feed our expressions for p obs , T , obs and T , obs intoEquation 2 from Seager & Mall´en-Ornelas (2003) to de-rive the observed impact parameter: b = (1 − p true B − / ) − sin ( T π/P )sin ( T π/P ) (1 + p true B − / ) − sin ( T π/P )sin ( T π/P ) . (A4)Let us assume that B = 1 i.e. no blend is present.Plugging equations Equation 1 into Equation A4 in thislimit yields lim B → b = b , (A5)as expected. Now consider that a blend source ispresent. Again feeding Equation 1 Equation A4 yields(without any approximation): b = B + p − √B (1 + p − b ) B . (A6)Since the above expression clearly scales with b true ,then we may find the maximum/minimum range of theabove by evaluating when b true → b true , min = 0 and b true → b true , max = 1 − p true : B + p B − p √B (cid:54) b (cid:54) B + p B − p true √B (A7)If we replace p true with p obs √ B then the RHS sim-plifies to b obs (cid:54) (1 − p − obs), displaying an analogousform the the boundary condition imparted on the trueimpact parameter. In the limit of no blending and ex-treme blending, Equation A7 gives:0 (cid:54) lim B→ b (cid:54) (1 − p true ) (cid:54) lim B→∞ b (cid:54) . (A8)Therefore, we find that 0 (cid:54) b obs (cid:54) (1 − p true ). Recallthat we have also showed that 0 (cid:54) b obs (cid:54) (1 − p obs ).Since both statements are true, one must take precedentover the other and since p obs < p true then the latter limitis the more constraining one i.e. 0 (cid:54) b obs (cid:54) (1 − p obs ) forall B (cid:62) and 0 (cid:54) b true (cid:54) (1 − p true ).Equation A6 may be re-written by replacing the p true terms with the observed values to give b = 1 + p − (cid:32) B p − b √B (cid:33) . (A9)The inverse of this expression is easily shown to be: b = 1 + B p − √B (1 − b ) − p . (A10) A3 Scaled semi-major axis bias
The scaled semi-major axis, ( a/R (cid:63) ), can also be derivedfrom the observed durations and ratio-of-radii via Equa-tion 3, to give: c (cid:13)000
The scaled semi-major axis, ( a/R (cid:63) ), can also be derivedfrom the observed durations and ratio-of-radii via Equa-tion 3, to give: c (cid:13)000 , 1–26 David M. Kipping ( a/R (cid:63) ) = (1 + p obs ) − b (1 − sin ( T π/P )sin ( T π/P ) . (A11)In the limit of no blend, then b obs → b true and p obs → p true , giving the expected result thatlim B→ ( a/R (cid:63) ) = ( a/R (cid:63) ) . (A12)However, in the case of a non-unity blend factor, wefind ( a/R (cid:63) ) = (cid:32) ( a/R (cid:63) ) − b (1 + p ) − b (cid:33)(cid:32) (1 + p ) − [( a/R (cid:63) ) − (1 + p ) ][ B + p − √ B (1 + p − b )] B [( a/R (cid:63) ) − b ] (cid:33) . (A13) A4 Mean stellar density bias
Finally, we come to the parameter of interest, the meanstellar density, ρ (cid:63), obs . Following the definition in Equa-tion 4, we have: ρ (cid:63), obs ≡ π ( a/R (cid:63) ) / GP . (A14)As before, for an unblended target star we recoverlim B→ ρ (cid:63), obs = ρ (cid:63), true . (A15)For blended planets, the equation is more compli-cated, particular when we make the substitution that( a/R (cid:63) ) = ( GP ρ (cid:63), true ) / (3 π ).By inspection of Equation A13, we found thatassuming ( a/R (cid:63) ) (cid:29) (1 + p ) (which also imparts( a/R (cid:63) ) (cid:29) b since b < (1 + p ) in order for a transitto occur) allows for significant reduction in the form ofthe expression for ( ρ (cid:63), obs /ρ (cid:63), true ) to:lim ( a/R (cid:63) ) (cid:29) (1+ p ) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = (cid:32) (1 + p ) − b (1 + p ) − b (cid:33) / (A16)Since we assume that the transit displays a flat bot-tom, then b (cid:46) p (cid:46) p ) (cid:46) a/R (cid:63) ) (cid:29)
4. By defini-tion, ( a/R (cid:63) ) > a/R (cid:63) ) (cid:29)
4, some very short-periodobjects such as Kepler-78b have ( a/R (cid:63) ) ∼ a/R (cid:63) ) may be estimated using P and ρ (cid:63), true and converting into typical units of measurewe determine that our approximation is valid for: (cid:16) P days (cid:17) / (cid:29) . (cid:16) ρ (cid:63), true g cm (cid:17) − / (A17)Under this condition then, one may re-write Equa-tion A16 in terms of the observables:lim ( a/R (cid:63) ) (cid:29) (1+ p ) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = (cid:32) (1 + √B p obs ) − b √B ((1 + p obs ) − b ) (cid:33) / . (A18) A5 Solving for the blend parameter
Since the observer directly determines p obs and b obs , thenEquation A18 suggests that one should be able to invertthe above and infer B . However, in doing so, we findthat one recovers a quadratic solution and both rootsare ostensibly plausible: B + , − = 14 p (cid:32) − p obs + (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / [(1 + p obs ) − b ] ± (cid:34)(cid:16) p obs (cid:104) (2 + p obs ) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / − (cid:105) + (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / (1 − b ) (cid:17) − p (1 − b ) (cid:35) / (cid:33) (A19)There are several analytic insights that can be madewith this expression. Firstly, at the extreme solution of( ρ (cid:63), obs /ρ (cid:63), true ) = 1, example plots of the functions (e.g.see Figure 1) show that these points correspond to themaximum and minimum in B -space: B (cid:62) lim ( ρ (cid:63), obs /ρ (cid:63), true ) → B − = 1 , (A20) B (cid:54) lim ( ρ (cid:63), obs /ρ (cid:63), true ) → B + = (1 − b ) p . (A21)We also note that two functions, B + and B − , meetat an apparent minimum in ( ρ (cid:63), obs /ρ (cid:63), true )-space. Thispoint is found by solving ∂ ( ρ (cid:63), obs /ρ (cid:63), true ) /∂B = 0 for B ,giving: lim (cid:0) ρ(cid:63), obs ρ(cid:63), true (cid:1) → (cid:0) ρ(cid:63), obs ρ(cid:63), true (cid:1) min B = 1 − b p , (A22)which may be used to determine the equivalent loca-tion in ( ρ (cid:63), obs /ρ (cid:63), true )-space, corresponding to the max-imum and minimum limits of said parameter: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) (cid:62) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) min = (cid:32) p obs (1 + (cid:112) − b )(1 + p obs ) − b (cid:33) / , (A23) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) (cid:54) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) max = 1 . (A24) c (cid:13) , 1–26 sterodensity Profiling Crucially then, a measurement of ( ρ (cid:63), obs ρ (cid:63), true ) below theminimum or above the maximum should not be possiblefor any degree of blending. Such a case therefore wouldmean that another AP effect is responsible for the devia-tion (which realistically can only be the photo-eccentriceffect) or the star hosting the eclipsing body does notpossess a true stellar density equal to ρ (cid:63), true .Finally, we note that the contours never cross thesmall-planet limit found by evaluating Equation A18 inthe limit p obs (cid:28) p obs (cid:28) (cid:32) lim ( a/R (cid:63) ) (cid:29) (1+ p ) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17)(cid:33) = B − / . (A25) A6 Allowed Range of B − Plotting some example functions in Figure 1 reveals thatthe B + solutions extend up to suspiciously high B . Thisissue can be phrased mathematically by computing thetrue value of p once one corrects for the blending factor,given by Equation A2. Since a fundamental assumptionof our work is that a flat-bottomed transit is observed,then we expect p true < (1 − b true ) at all times and sincethe minimum value of b true is zero then the maximumlimit is p true <
1. By this criteria and inspection of thecontours in Figure 1, we note that there appear to besome apparently forbidden p true values along the B + con-tour with B reaching ∼ .Before exploring the very high blend factors pro-duced by the B + root, we first evaluate the maximumpossible p true value along the B − contour, which oc-curs at the point where the B − meets B + i.e. when (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) → (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) min :lim (cid:0) ρ(cid:63), obs ρ(cid:63), true (cid:1) → (cid:0) ρ(cid:63), obs ρ(cid:63), true (cid:1) min p true = (cid:113) − b (A26)We have already derived an expression for b obs ear-lier in Equation A9, which scales with b true . It was shownearlier than 0 (cid:54) b obs (cid:54) (1 − p obs ) for the allowed pa-rameter range considered in this study. This means that p true < (cid:112) − (1 − p obs ) (cid:54) p obs (cid:54)
1. This there-fore proves that all loci along the B − contour reside inunforbidden parameter space. A7 Allowed Range of B + It is easy to show that at least some of the loci alongthe B + contour produce p true > B + found when( ρ (cid:63), obs /ρ (cid:63), true ) →
1, as mentioned earlier:lim ( ρ (cid:63), obs /ρ (cid:63), true ) → B + = (1 − b ) p . (A27)Requiring p true < B < p − whichmeans that in order for the above satisfy this we require (1 − b ) < p obs . However, since b obs < − p obs as shownearlier in Appendix A2, then this condition can never bein effect. Therefore, there is no doubt that B + at leastpartially samples forbidden parameter space.We may actually solve for the point along B + whenthis breakdown occurs. This must occurs when B + = p − since we require p true < B < p − at alltimes. Solving this expression for ( ρ (cid:63), obs /ρ (cid:63), true ) yields aquadratic equation with two roots. The first root has thesolution: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = (cid:32) b p obs (1 + p obs ) − b (cid:33) / . (A28)This may be compared to the minimum allowedvalue of ( ρ (cid:63), obs /ρ (cid:63), true ) derived earlier in Equation A24,meaning that we require: (cid:32) b p obs (1 + p obs ) − b (cid:33) / > (cid:32) p obs (1 + (cid:112) − b )(1 + p obs ) − b (cid:33) / , ⇒ b ≯ (cid:113) − b ) ∀ < b obs < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) B + , max = (cid:32) (4 − b ) p obs (1 + p obs ) − b (cid:33) / , (A30)which does satisfy the condition of being greaterthan the minimum estimate in Equation A24 for all b obs >
0. This maximum limit is marked with gray cir-cles on the example plots shown in Figure 1. Therefore, B + produces is a valid solution when we have: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) min < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) < (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) B + , crit . (A31)A measurement of the density in this range meansthat the inverse solution for B has two roots. There-fore, one should expect a bi-modal posterior distribu-tion for B when using SAP in such cases, provided theprior range in B is allowed to explore to high blend fac-tors. We also note that in the the limit of b → (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) min = (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) B + , crit meaning the B + solutionis always forbidden in such a case. This is also evidentin the top-left panel of Figure 1 where the two curvescorresponding to these limits overlap. APPENDIX B: VALID RANGE FOR THEANALYTIC PHOTO-ECCENTRIC EFFECTB1 The Transit Duration Equation
In this paper, we assume that the observed stellar densityis affected by the photo-eccentric effect via a simple ana-lytic formula. In this section, we investigate under what c (cid:13)000
In this paper, we assume that the observed stellar densityis affected by the photo-eccentric effect via a simple ana-lytic formula. In this section, we investigate under what c (cid:13)000 , 1–26 David M. Kipping conditions this simple formula is actually a valid sincethere appears to be no previous efforts to quantify thevalidity of this crucial assumption. The observed stellardensity is assumed to behave as (Kipping 2010a): (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = Ψ , (B1)where we defineΨ ≡ (1 − e sin ω ) (1 − e ) / . (B2)Inferences about the eccentricity of a system madeusing the above expressions are defined here as exploit-ing the analytic photo-eccentric. These expressions aretaken from Kipping (2010a) but we note that many otherauthors have used this function for modeling the photo-eccentric effect (Winn 2010; Carter 2011; Kipping et al.2012; Dawson & Johnson 2012). Given the widespreaduse of this expression, it is crucial to understand the lim-its of the equation in question. The expressions above arederived by setting T , obs and T , obs to that expected fora planet with orbital eccentricity, e , and argument of pe-riastron, ω . To date, there is no known exact analyticexpression for the duration of a transit on an eccentricorbit but Kipping (2010a) derived an approximate ex-pression, provided by Equation 15 of that work: T = Pπ (cid:37) c √ − e sin − (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b , (B3)where Kipping (2010a) define (cid:37) c ≡ − e e sin ω . (B4)Kipping (2010a) demonstrate that Equation B3 isan excellent approximation to the true transit dura-tion (which can be computed more laboriously via themethod described in Kipping 2008). As demonstrated inKipping (2010a), these approximate expressions becomemost erroneous when ( a/R (cid:63) ) is small. However, even at( a/R (cid:63) ) = 5, the expression performs better than 1% ac-curacy across the vast majority of parameter space andthe paper finds an impressive average accuracy of < . | e sin ω | < . | e cos ω | < .
85. Compared to theother assumptions made in deriving Equation B1, whichwe will shortly discuss, Equation B3 is unlikely to everbe the bottleneck in accuracy.Although Kipping (2010a) spent great effort explor-ing the accuracy of Equation B3, no effort is spent on theaccuracy of the most relevant equation for the analyticphoto-eccentric effect i.e. Equation B1. The reason forthis is quite simply that the photo-eccentric effect hadnot been envisaged at this time and so the importanceof Equation B1 was not realized. Therefore, we devotethis section to addressing this important question.
B2 Accuracy of the Impact ParameterApproximation
As with other asterodensity effects, not only is ( a/R (cid:63) ) obs (and thus ρ (cid:63), obs ) deviant from the truth, but also the ob-served impact parameter, b obs , is deviant. Using Equa-tion B3 and the original Seager & Mall´en-Ornelas (2003)equations, Kipping (2010a) (see Equation 33) showedthat (without any approximations): b = 1 + p + 2 p (cid:32) sin [ (cid:37) c √ − e sin − ( √ (1 − p ) − b a R (cid:37) c sin i )] + sin [ (cid:37) c √ − e sin − ( √ (1+ p ) − b a R (cid:37) c sin i )]sin [ (cid:37) c √ − e sin − ( √ (1 − p ) − b a R (cid:37) c sin i )] − sin [ (cid:37) c √ − e sin − ( √ (1+ p ) − b a R (cid:37) c sin i )] (cid:33) . (B5)Kipping (2010a) briefly remark that making a small-angle approximation in the trigonometric functions al-lows one to simplify the above to b = b . However,what remains unclear is exactly under what conditions isa small-angle approximation valid? The small-angle ap-proximation is actually implemented four times in total,two of which correspond to sin − x (cid:39) x and two of whichcorrespond to sin x (cid:39) x . Let us begin by inspecting thevalidity of the inverse sine approximation. B2.1 Accuracy of the inverse sine small-angleapproximation
The inverse sine approximation in question is fully ex-pressed as:sin − (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b (cid:39) (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b . (B6)It is trivial to show that the (cid:37) c term has two extremaat ω = π/ ω = 3 π/ [1] (1 ± p ) → (1 + p ) and (cid:37) c → lim ω → π/ (cid:37) c [2] (1 ± p ) → (1 + p ) and (cid:37) c → lim ω → π/ (cid:37) c [3] (1 ± p ) → (1 − p ) and (cid:37) c → lim ω → π/ (cid:37) c [4] (1 ± p ) → (1 − p ) and (cid:37) c → lim ω → π/ (cid:37) c In each case, the remaining variables are ( a/R (cid:63) ), b , e and p . Let us proceed by finding the maximum ofthe inverse sine function’s argument in all four cases.We demonstrate this by Monte Carlo experiment wherewe draw random uniform variates for 0 < p < < b < (1 − p ) and 0 < e < e max . For each realization, weplot the inverse sine argument as a function of the onlyremaining dependent variable, ( a/R (cid:63) ). We make 1000plots for each of the four cases and in each case we de-termine the maximum value of the inverse sine function’sargument, with respect to ( a/R (cid:63) ) and e max . In practice c (cid:13) , 1–26 sterodensity Profiling this is done by both varying the experiments for severaldifferent e max values and taking the derivatives of theinverse sine function’s argument. In all Monte Carlo ex-periments, we enforce the condition that ( a/R (cid:63) ) > (1 − e )to avoid the planet colliding into the star. Figure B1 dis-plays our results when we arbitarily choose e max = 0 . a/R (cid:63) ) and e max . From thesefour maxima functions, one may use the maximum of these to demonstrate that: (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b (cid:54) a/R (cid:63) )(1 − e ) ∀ (0 (cid:54) p < (cid:54) b < − p ); (0 (cid:54) ω < π ); (0 (cid:54) e < . (B7)Armed with the above, one may now answer thequestion as to what range of orbits the small-angle in-verse sine approximation is valid. The Maclaurin seriesexpansion of the inverse sine function may be expressedas: sin − x = x + x O [ x ] . (B8)Therefore, the approximation that sin − x (cid:39) x isvalid when ( x / (cid:28) x i.e. when ( x / (cid:28)
1. Usingour maximum expression for the inverse sine argumentin Equation B7, the small-angle approximation is nowvalid for:
Condition A ( a/R (cid:63) ) (cid:29)
23 1(1 − e ) . (B9) B2.2 Accuracy of the sine small-angle approximation
Let us assume that Condition A is valid so that:sin (cid:32) (cid:37) c √ − e sin − (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b (cid:33) (cid:39) sin (cid:32) (cid:37) c √ − e (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b (cid:33) . (B10)Next, we need to investigate when the small-angleapproximation for the sine function is valid i.e. whensin (cid:32) (cid:37) c √ − e (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b (cid:33) (cid:39) (cid:32) (cid:37) c √ − e (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b (cid:33) . (B11)As with the investigation of the inverse sine function,we will consider the four extreme cases of: [1] (1 ± p ) → (1 + p ) and (cid:37) c → lim ω → π/ (cid:37) c [2] (1 ± p ) → (1 + p ) and (cid:37) c → lim ω → π/ (cid:37) c [3] (1 ± p ) → (1 − p ) and (cid:37) c → lim ω → π/ (cid:37) c [4] (1 ± p ) → (1 − p ) and (cid:37) c → lim ω → π/ (cid:37) c As before, we seek to determine the maximum ofthe sine function’s argument with respect to ( a/R (cid:63) ) and e max by Monte Carlo experiments and analysis of thedifferentials. Generating random p , b and e values via thesame method used earlier, we determine upper limits foreach of the four cases, shown in Figure B2.After conducting this analysis, we are able to derivefunctional upper limits on the the sine function’s argu-ment with respect to ( a/R (cid:63) ) and e max . From these fourmaxima functions, one may use the maximum of these to demonstrate that: (cid:37) c √ − e (cid:115) (1 ± p ) − b ( a/R (cid:63) ) (cid:37) c − b (cid:54) a/R (cid:63) ) (cid:16) e − e (cid:17) / ∀ (0 (cid:54) p < (cid:54) b < − p ); (0 (cid:54) ω < π ); (0 (cid:54) e < . (B12)Armed with the above, one may now answer thequestion as to what range of orbits the small-angle sineapproximation is valid. The Maclaurin series expansionof the inverse sine function may be expressed as:sin x = x − x O [ x ] . (B13)Therefore, the approximation that sin x (cid:39) x is validwhen ( x / (cid:28) x i.e. when ( x / (cid:28)
1. Using our maxi-mum expression for the sine argument in Equation B12,the small-angle approximation is now valid for:
Condition B ( a/R (cid:63) ) (cid:29)
23 (1 + e ) (1 − e ) . (B14) B2.3 Summary
We have now derived the conditions under which thesmall-angle inverse sine approximation (Equation B9)and the small-angle sine approximation (Equation B14)are valid. It is easily shown that Condition B always leadsto a harder constraint on ( a/R (cid:63) ), meaning that Condi-tion A is superfluous. Applying the small-angle approxi-mations to Equation B5 elegantly recovers b , as Kipping(2010a) stated. However, the actual limit of this approx-imation is now quantified as:lim ( a/R (cid:63) ) (cid:29) [2(1+ e ) ] / [3(1 − e ) ] b obs = b. (B15) c (cid:13)000
We have now derived the conditions under which thesmall-angle inverse sine approximation (Equation B9)and the small-angle sine approximation (Equation B14)are valid. It is easily shown that Condition B always leadsto a harder constraint on ( a/R (cid:63) ), meaning that Condi-tion A is superfluous. Applying the small-angle approxi-mations to Equation B5 elegantly recovers b , as Kipping(2010a) stated. However, the actual limit of this approx-imation is now quantified as:lim ( a/R (cid:63) ) (cid:29) [2(1+ e ) ] / [3(1 − e ) ] b obs = b. (B15) c (cid:13)000 , 1–26 David M. Kipping
Figure B1.
Small-angle inverse sine approximation investigation:
On the y -axis we plot the four extreme possible argumentsto the inverse sine functions present in Equation B5, with respect to ( a/R (cid:63) ) on the x -axis. Each panel shows 1000 randomrealizations for p , b and e , where the RGB-colouring is given by { R , G , B } = { p, b, e } . For each panel, we show the maximumallowed value of the function in black-dashed. Simulations produced using e max = 0 . , but the upper limits are valid for all (cid:54) e max < . B3 Accuracy of the Density Approximation
With the valid range for assuming b obs = b now resolved,we may proceed to finally broach the question as to whenEquation B1 is valid i.e. when the analytic model for thephoto-eccentric effect can be employed. The stellar den-sity is trivially computed from ( a/R (cid:63) ) and so it is morepertinent to phrase the question as to what is ( a/R (cid:63) ) obs ?Kipping (2010a) (Equation 35) showed that (without anyapproximation):lim b obs → b ( a/R (cid:63) ) = b +[(1 + p ) − b ] csc (cid:34) (cid:37) c √ − e sin − (cid:115) (1 + p ) − b ( a/R (cid:63) (cid:37) c − b (cid:35) . (B16)At this point Kipping (2010a) again invoke an in-verse sine and sine small-angle approximation to sim-plify the above. However, making these approximationsare equivalent to cases [1] & [2] of the inverse sine approx-imation and cases [1] & [2] of the sine function approxi- mation made earlier in this section. Therefore, since wehave already assumed Condition B (Equation B14) is ineffect in order to approximate b obs = b , then it necessar-ily follows that both of these small-angle approximationsmust also be valid. Making these approximations allowsfor significant simplification, yielding the same result asEquation 36 of Kipping (2010a):lim ( a/R (cid:63) ) (cid:29) [2(1+ e ) ] / [3(1 − e ) ] ( a/R (cid:63) ) obs = ( a/R (cid:63) ) (cid:115) (cid:37) c cos i + (1 − e ) sin i(cid:37) c . (B17)The final approximation made in Kipping (2010a),which ultimately yields the photo-eccentric Ψ equation,is that the system is nearly coplanar. This essentiallymeans that we adopt cos i = 0 and sin i = 1 in the aboveand doing so recovers Equation B1. Explicitly though,the assumption may be expressed as: c (cid:13) , 1–26 sterodensity Profiling Figure B2.
Small-angle sine approximation investigation:
On the y -axis we plot the four extreme possible arguments to thesine functions present in Equation B5, with respect to ( a/R (cid:63) ) on the x -axis. Each panel shows 1000 random realizations for p , b and e , where the RGB-colouring is given by { R , G , B } = { p, b, e } . For each panel, we show the maximum allowed value of thefunction in black-dashed. Simulations produced using e max = 0 . , but the upper limits are valid for all (cid:54) e max < . − e (cid:37) c sin i (cid:29) cos i. (B18)Replacing sin i with (1 − cos i ) and then replacingcos i with b/ [( a/R (cid:63) ) (cid:37) c ] gives:( a/R (cid:63) ) (cid:29) (cid:37) c + 1 − e − e b (cid:37) c . (B19)The function on the RHS depends upon b , e and ω and implicitly p (since 0 < b < − p ). As with earlier,we seek a simple form for the maximum of the term onRHS by Monte Carlo experiment. In Figure B3, we show1000 random realizations of this function plotted withrespect to b , drawing uniform variates for 0 (cid:54) p < (cid:54) e < e max and 0 (cid:54) ω < π . The exercise reveals thatthe function is bounded by (cid:37) c + 1 − e − e b (cid:37) c (cid:54) b (cid:16) − e ) + 2(1 + e ) − (cid:17) . (B20)We may now use the above and evaluate it when b = 1, which maximizes the limit, to give: Condition C ( a/R (cid:63) ) (cid:29) (cid:16) − e ) + 2(1 + e ) − (cid:17) (B21)Since have assumed Condition B already, it is worthcomparing the above to Equation B14. Plotting the twofunctions out in Figure B4 one sees that unlike the casewhere we compared Conditions A & B, one function doesnot always dominate over the other. However, the pointof intersection occurs for the constraint that ( a/R (cid:63) ) (cid:29) .
2, after which point Condition B dominates. Therefore,provided we are willing to assume the quite reasonablescenario that ( a/R (cid:63) ) (cid:29) a/R (cid:63) ) (cid:29) (cid:32) e − e (cid:33) , (B22)or equivalently this may be re-expressed in physi- c (cid:13)000
2, after which point Condition B dominates. Therefore,provided we are willing to assume the quite reasonablescenario that ( a/R (cid:63) ) (cid:29) a/R (cid:63) ) (cid:29) (cid:32) e − e (cid:33) , (B22)or equivalently this may be re-expressed in physi- c (cid:13)000 , 1–26 David M. Kipping
Figure B3.
Coplanar approximation investigation:
Monte Carlo realizations for the constraint on the ( a/R (cid:63) ) function, expressed on the y -axis and given in Equation B19,with respect to b . We show 1000 random realizations for thefunction by drawing random uniform variates for p , ω and p , which respectively define the RGB-colouring scheme. Theblack-dashed lined describes the observed upper limit. Simu-lations produced using e max = 0 . , but the upper limits arevalid for all (cid:54) e max < . Figure B4.
Comparison of conditions B & C:
Here weplot the RHS of Equations B14&B21 in order to visualizewhich of the two conditions dominates. Under the reasonableassumption that ( a/R (cid:63) ) (cid:29) , then Condition B can be seento dominate and thus we dub this the analytic photo-eccentriccondition. cal dimensions by re-writing ( a/R (cid:63) ) in terms of the truestellar density (cid:32) P days (cid:33) / (cid:29) . (cid:32) ρ (cid:63), true g cm − (cid:33) − / (cid:32) e − e (cid:33) . (B23)Adopting the analytic photo-eccentric conditionmeans one may now re-write Equation B1 as lim ( a/R (cid:63) ) (cid:29) [2(1+ e ) ] / [3(1 − e ) ] (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = Ψ . (B24) APPENDIX C: DERIVATION OF THEMINIMUM ECCENTRICITY EQUATION
In this work, we have presented a new expression for theminimum eccentricity of an exoplanet (Equation 39), as afunction of the observed and true stellar densities ( ρ (cid:63), obs and ρ (cid:63), true respectively). Here, we present a derivation ofthis equation. As with the other derivations in this work,we ignore other effects (e.g. photo-blend, photo-mass,etc) during the course of this derivation and assumethe analytic photo-eccentric condition (Equation B23)is satisfied. Accordingly, the ratio of the observed stellardensity to the true stellar density follows the expression(Kipping 2010a): (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = Ψ (C1)where Ψ = (1 + e sin ω ) (1 − e ) / . (C2)( ρ (cid:63), obs /ρ (cid:63), true ) can therefore be seen to be a func-tion of two parameters, e and ω , meaning that we haveone observable and two unknowns. Progress can be madeon this under-constrained problem by considering the ex-trema (i.e. the minima/maxima) of the expression. Wewill proceed by taking the extrema with respect to ω ,which is easily achieved by computing the derivative withrespect to ω . Solving ∂ ( ρ (cid:63), obs /ρ (cid:63), true ) /∂ω = 0 for ω un-der the condition that 0 (cid:54) e < ω = π/ ω = 3 π/ ω → π/ (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = (cid:16) − e (cid:17) − / , (C3)lim ω → π/ (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = (cid:16) − e (cid:17) / . (C4)Let us solve the above expressions so that e is thesubject: lim ω → π/ e = (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / + 1 , (C5)andlim ω → π/ e = (1 − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / )(1 − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / + (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / )1 + (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) . (C6)If ( ρ (cid:63), obs /ρ (cid:63), true ) >
1, then Equation C5 yields apositive eccentricity, otherwise it is negative. A negative c (cid:13) , 1–26 sterodensity Profiling eccentricity of course has no meaning and this can beexplained by the fact that if ω = π/ ρ (cid:63), obs /ρ (cid:63), true ) < ρ (cid:63), obs /ρ (cid:63), true ) < ρ (cid:63), obs /ρ (cid:63), true ) > ω = 3 π/ ρ (cid:63), obs /ρ (cid:63), true ) >
1, then we should use Equation C5and if we have ( ρ (cid:63), obs /ρ (cid:63), true ) < ρ (cid:63), obs /ρ (cid:63), true ) > ρ (cid:63), obs /ρ (cid:63), true ) < e min = (cid:16) lim ω → π/ e (cid:17) H (cid:104)(cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) − (cid:105) + (cid:16) lim ω → π/ e (cid:17) H (cid:104) − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1)(cid:105) , (C7)which we evaluate to be e min = (cid:32) (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / + 1 (cid:33) H (cid:104)(cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) − (cid:105) + (cid:32) (1 − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / )(1 − (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / + (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) / )1 + (cid:0) ρ (cid:63), obs ρ (cid:63), true (cid:1) (cid:33) H (cid:104) − (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17)(cid:105) , (C8)Note that e min is purely a function of ( ρ (cid:63), obs /ρ (cid:63), true )and no other terms. It is therefore possible to analyticallycalculate the uncertainty on e min using quadrature: σ e min = 43 (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) − / (cid:32) (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) / (cid:33) − σ ρ (cid:63), obs /ρ (cid:63), true , (C9)where σ e min and σ ρ (cid:63), obs /ρ (cid:63), true are the uncertaintieson the minimum eccentricity and ( ρ (cid:63), obs /ρ (cid:63), true ) observ-able respectively. APPENDIX D: DERIVATION OF THEPHOTO-TIMING EFFECT
Consider N (cid:29) b . Since b isinversely correlated to ( a/R (cid:63) ) and thus ρ (cid:63) , one shouldanticipate that unaccounted for TTVs will produce anartificially lower ρ (cid:63) value. This may be formally provedhere by considering the effect on the contact points andfollowing the method outlined in Kipping (2010b). Fora peak-to-peak TTV amplitude of 2 A TTV , the contactpoints of the composite signal appear shifted by: t I, obs = t I, true − A TTV , (D1) t II, obs = t II, true + A TTV , (D2) t III, obs = t III, true − A TTV , (D3) t IV, obs = t IV, true + A TTV . (D4)(D5)Together, these change the apparent transit dura-tions, T and T , to: T , obs = T , true − A TTV , (D6) T , obs = T , true + 2 A TTV . (D7)Extreme scenarios can cause T , obs < p obs = p true .In order to compute the deviation in ( ρ (cid:63), obs /ρ (cid:63), true )from unity due the photo-timing effect, one may fol-low the methodology outlined in § ρ (cid:63), obs /ρ (cid:63), true )and rather than formally stating the full equation (re-quiring many lines), meaningful insights may be drawnby plotting the resulting function for various impact pa-rameters.In Figure D1, we plot the ratio ( ρ (cid:63), obs /ρ (cid:63), true ) as afunction of ( A TTV /P ) for several iso- b contours. The plotreveals that the maximal error in ( ρ (cid:63), obs /ρ (cid:63), true ) occursfor b = 0 and so we may continue by focussing our effortson this case and interpreting it as the maximal deviation.We find that for ( A TTV /P ) (cid:39) − the ( ρ (cid:63), obs /ρ (cid:63), true )term is deviant by 1%. c (cid:13)000
Consider N (cid:29) b . Since b isinversely correlated to ( a/R (cid:63) ) and thus ρ (cid:63) , one shouldanticipate that unaccounted for TTVs will produce anartificially lower ρ (cid:63) value. This may be formally provedhere by considering the effect on the contact points andfollowing the method outlined in Kipping (2010b). Fora peak-to-peak TTV amplitude of 2 A TTV , the contactpoints of the composite signal appear shifted by: t I, obs = t I, true − A TTV , (D1) t II, obs = t II, true + A TTV , (D2) t III, obs = t III, true − A TTV , (D3) t IV, obs = t IV, true + A TTV . (D4)(D5)Together, these change the apparent transit dura-tions, T and T , to: T , obs = T , true − A TTV , (D6) T , obs = T , true + 2 A TTV . (D7)Extreme scenarios can cause T , obs < p obs = p true .In order to compute the deviation in ( ρ (cid:63), obs /ρ (cid:63), true )from unity due the photo-timing effect, one may fol-low the methodology outlined in § ρ (cid:63), obs /ρ (cid:63), true )and rather than formally stating the full equation (re-quiring many lines), meaningful insights may be drawnby plotting the resulting function for various impact pa-rameters.In Figure D1, we plot the ratio ( ρ (cid:63), obs /ρ (cid:63), true ) as afunction of ( A TTV /P ) for several iso- b contours. The plotreveals that the maximal error in ( ρ (cid:63), obs /ρ (cid:63), true ) occursfor b = 0 and so we may continue by focussing our effortson this case and interpreting it as the maximal deviation.We find that for ( A TTV /P ) (cid:39) − the ( ρ (cid:63), obs /ρ (cid:63), true )term is deviant by 1%. c (cid:13)000 , 1–26 David M. Kipping lim b → (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = (cid:16) πGP (cid:17) × csc (cid:104) nA TTV + (1 + p )( a/R (cid:63) ) − (cid:105)(cid:34) (1 + p ) + cos[ nA TTV + (1 + p )( a/R (cid:63) ) − ] × (cid:32) (1 + p ) sin [ nA TTV − (1 − p )( a/R (cid:63) ) − ]sin [ nA TTV + (1 + p )( a/R (cid:63) ) − ] − (1 − p ) (cid:33)(cid:32) − sin [ nA TTV − (1 − p )( a/R (cid:63) ) − ]sin [ nA TTV + (1 + p )( a/R (cid:63) ) − ] (cid:33) − (cid:35) / , (D8)where n = 2 π/P . Making small-angle approxima-tions of the various trigonometric terms, allows for con-siderable simplification of this equation:lim b → (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) (cid:39) (cid:32) pp + nA TTV ( a/R (cid:63) ) true (cid:33) / (D9)Similar small-angle approximations have been madepreviously in this work in Appendix B, where we de-rived the exact parameter range of the approximation’svalidity. In the case of the photo-timing effect derivationpresented here, we need only concern ourselves with the e → A TTV (cid:28) T for the exact same approxima-tion to be valid. Let us invoke this reasonable assump-tion since any large TTVs which break this conditionshould be easily detected and compensated for and thephoto-timing effect concerns itself with clandestine tim-ing variations. In Appendix B, we found that two condi-tions were required for the small-angle approximations;conditions B & C, given by Equations B14 & B21. Usingthose same expressions, but setting e = 0 as appropriatefor the photo-timing derivation being considered here, wefind that such an approximation is generally valid if:( a/R (cid:63) ) (cid:29) , (D10)2 A TTV (cid:28) T . (D11)which can be considered to be true for the vastmajority of orbital configurations. As visible from Fig-ure D1, the approximation given by Equation D9 doesan excellent job of reproducing the behavior of the exactsolution for b = 0. This equation is also highly practicalin estimating the error in ( ρ (cid:63), obs /ρ (cid:63), true ) when some up-per limit on the TTVs has been derived, since the b = 0limit is the most conservative case and in general the de-rived b will over-estimated and thus unreliable anywaydue to the TTV smearing. This TTV smearing imposesa fundamental limit on the precision at which one canmeasure ( ρ (cid:63), obs /ρ (cid:63), true ). To make Equation D9 of even greater practical valueto observers, it is useful to replace ( a/R (cid:63) ) true with ρ (cid:63), true since this parameter is more directly inferred from anindependent measure of the star. Further, in the case ofno detected TTVs, the A TTV may be replaced with theupper limit on the TTV amplitude and the LHS may beinterpretted as the uncertainty in ( ρ (cid:63), obs /ρ (cid:63), true ): σ max( ρ (cid:63), obs /ρ (cid:63), true ) = 1 − (cid:32) π / / G / ρ / (cid:63), true pP / σ A TTV (cid:33) − / , (D12)where σ ( ρ (cid:63), obs /ρ (cid:63), true ) (cid:54) σ max( ρ (cid:63), obs /ρ (cid:63), true ) (D13)and σ A TTV is the 1 σ upper limit on the presence ofTTVs. For N (cid:29) σ τ , one expects thestandard deviation of the TTV points in the absence ofa signal to be σ τ . The uncertainty on this prediction(i.e. the standard deviation of the standard deviation) isgiven by σ τ / (cid:112) N −
1) assuming normally distributederrors. The 1 σ maximum standard deviation can then becompared to that expected from an embedded sinusoidwithin the data which could cause a standard deviationof (cid:112) σ τ + ( A / (cid:112) σ τ + ( A /
2) = σ τ + σ τ / (cid:112) N − . (D14)Solving for A gives σ A TTV as σ A TTV = σ τ (cid:115) N − (cid:114) N − , (D15)lim N (cid:29) σ A TTV (cid:39) σ τ (cid:16) N (cid:17) / . (D16)We may now plug the above result into Equa-tion D12. Further, we take an approximate estimate ofthe ρ (cid:63), true term on the RHS of Equation D12 to be equalto ρ (cid:63), obs . Finally, we assume that the fractional error ismuch less than unity to simplify the expression to σ max( ρ (cid:63), obs /ρ (cid:63), true ) (cid:39) . G / ρ / (cid:63), obs pP / σ τ N / . (D17)In a Taylor expansion of [1 − (1 + x ) / ], we require x (cid:28) . (cid:32) P days (cid:33) / (cid:29) p (cid:32) A TTV seconds (cid:33)(cid:32) ρ (cid:63), true g cm − (cid:33) − / . (D18) c (cid:13) , 1–26 sterodensity Profiling Figure D1.
The effect of unaccounted for transit timing vari-ations (x-axis) on the observed mean stellar density (y-axis)from a composite transit light curve. From red to blue we showiso- b contours of b = 0 . , . , . , . & . respectively. Theblack-dashed line shows the result of our approximate expres-sion in the b = 0 limit (Equation D9). Realizations computedusing P = 10 days, ρ (cid:63) = ρ (cid:12) and p = 0 . . APPENDIX E: DERIVATION OF THEPHOTO-DURATION EFFECT
Consider a planet undergoing periodic, low-amplitudevelocity-induced transit duration variations (TDV-Vs).By periodically increasing/decreasing the velocity of aplanet, one expects the transit duration to scale inversely.In Figure 4, the effect is illustrated on the composite lightcurve.The outcome of unaccounted TDV-Vs is similar tothat of unaccounted TTVs. Namely, the first and fourthcontacts are pulled outwards and the second and thirdcontacts are pulled inwards. Thus one should expect un-accounted TDV-Vs to cause one to underestimate thestellar density, like TTVs.In what follows we consider the effect of a sinusoidalvelocity variation via v ( t ) = v [1 − A TDV sin(2 πt/P
TDV )] . (E1)However, it is important to note the derivation isgeneral for any periodic waveform and in this sense ourmodels defines A TDV as half of the peak-to-peak velocityvariation amplitude. Since T is linearly inversely pro-portional to the velocity, v , then we have: T ( t ) = T , [1 − A TDV sin(2 πt/P
TDV )] − . (E2)If A TDV (cid:28)
1, then we have: T ( t ) (cid:39) T , [1 + A TDV sin(2 πt/P
TDV )] (E3)In such a case, one can show that the compositecontact points are shifted by t I, obs = t I, true − A TDV T , , (E4) t II, obs = t II, true + A TDV T , , (E5) t III, obs = t III, true − A TDV T , , (E6) t IV, obs = t IV, true + A TDV T , . (E7)(E8)Together, these change the observed transit dura-tions, T and T , to: T , obs = T , true − A TDV T , true , (E9) T , obs = T , true + 2 A TDV T , true . (E10)One may now proceed to derive the effect on ρ (cid:63), obs as we did before for the photo-timing effect. However,unlike the photo-timing effect, we find that a simple formof the equation is possible for all b values, given by: (cid:16) ρ (cid:63), obs ρ (cid:63), true (cid:17) = (cid:32) ( a/R (cid:63) ) p + 4 A b p + 2 A TDV [(1 − p ) − b (1 + p )]( a/R (cid:63) ) [ p + 4 A p + 2 A TDV (1 + p − b )] (cid:33) / , (E11)where ( a/R (cid:63) ) is ( a/R (cid:63) ) true and can be estimated as[( GP ρ (cid:63) ) / (3 π )] / . As with the previous derivations, theabove required making similar small-angle approxima-tions to those made in Appendix B. These approxima-tions are valid here too under the already made assump-tion that A TDV (cid:28)
1, meaning we assume:( a/R (cid:63) ) (cid:29) , (E12) A TDV (cid:28) . (E13)This paper has been typeset from a TEX/ L A TEX file pre-pared by the author. c (cid:13)000