Transiting Extrasolar Planet with a Companion: Effects of Orbital Eccentricity and Inclination
aa r X i v : . [ a s t r o - ph . E P ] J u l Transiting Extrasolar Planet with a Companion:Effects of Orbital Eccentricity and Inclination
Masanao
Sato , and Hideki
Asada
Faculty of Science and Technology, Hirosaki University, Hirosaki, Aomori 036-8561 (Received ; accepted )
Abstract
Continuing work initiated in an earlier publication [Sato and Asada, PASJ, 61, L29(2009)], we consider light curves influenced by the orbital inclination and eccentricityof a companion in orbit around a transiting extrasolar planet (in a planet-satellitesystem or a hypothetical true binary). We show that the semimajor axis, eccentricityand inclination angle of a ‘moon’ orbit around the host planet can be determined bytransit method alone. For this purpose, we present a formulation for the parameterdeterminations in a small-eccentricity approximation as well as in the exact form. Asa result, the semimajor axis is expressed in terms of observables such as brightnesschanges, transit durations and intervals in light curves. We discuss also a narrowregion of parameters that produce a mutual transit by an extrasolar satellite.
Key words: techniques: photometric — eclipses — occultations — planets andsatellites: general — stars: planetary systems
1. Introduction
It is of general interest to discover a second Earth-Moon system. Detections of suchan extrasolar planet with a satellite (or hypothetical binary planet systems that do not existin the Solar System) and probing the nature of such objects will bring important informationto planet (and satellite) formation theory (e.g., Williams et al. 1997, Jewitt and Sheppard2005, Canup and Ward 2006, Jewitt and Haghighipour 2007). If a giant planet with a (perhapsEarth-size) rocky satellite were located at a certain distance from their host star, the satellitemay be habitable and show vegetation, though these issues are out of the scope of this paper.It is not clear whether the IAU definition for planets in the Solar System can be appliedto extrasolar planets as it is. The IAU definition in 2006 is as follows: A planet is a celestialbody that (a) is in orbit around the Sun, (b) has sufficient mass so that it assumes a hydrostaticequilibrium (nearly round) shape, and (c) has cleared the neighborhood around its orbit.Regarding (c), the Earth can be called a planet, mostly because the common center of1ass (COM) of the Earth-moon system is below the surface of the Earth. On the other hand,the COM of the Pluto-Charon system is located above the surfaces of these objects. Therefore,it is interesting to determine the COM position of a planet-companion system. In order todetermine it, we have to know the true (not apparent) distance between the two objects. Forthis reason, Sato and Asada (2009) considered extrasolar mutual transits, as a complementarymethod of measuring not only the radii of two transiting objects but also their separation(See Sato and Asada 2009 also on detection probabilities of extrasolar mutual transits and apossible limit by Kepler Mission). As a particular case, a short separation binary, which hasa rapidly orbiting companion, gives us a unique opportunity to measure the true separation ofthe binary, whereas a long separation one gives us only the apparent separation. Their work isvery limited, however, in the sense that they assume circular orbits and also coplanar orbits as I = 90 degrees for both planet and moon. Clearly it is important to take account of the orbitalinclination and eccentricity. The main purpose of this paper is to study effects of the orbitalinclination and eccentricity of a companion on extrasolar mutual transits.Since the first detection of a transiting extrasolar planet (Charbonneau et al. 2000),photometric techniques have been successful (e.g., Deming et al. 2005 for probing atmosphere,Ohta et al. 2005, Winn et al. 2005, Gaudi & Winn 2007, Narita et al. 2007, 2008 for measuringspin-orbit alignment angle). In addition to COROT , Kepler is monitoring about 10 stars withexpected 20 ppm (= 2 × − ) photometric differential sensitivity for stars of V=12. This willmarginally enable the detection of a moon-size object. In fact, COROT detected a transitingsuper-Earth (Leger et al. 2009, Queloz et al. 2009).Sartoretti and Schneider (1999) first suggested a photometric detection of extrasolarsatellites. Cabrera and Schneider (2007) developed a method based on the imaging of a planet-companion as an unresolved system (but resolved from its host star) by using planet-companionmutual transits and mutual shadows. As an alternative method, timing offsets for a singleeclipse have been investigated for eclipsing binary stars as a perturbation of transiting planetsaround the center of mass in the presence of the third body (Deeg et al. 1998, 2000, Doyle et al.2000). It has been recently extended toward detecting ‘exomoons’ (Szab´o et al. 2006, Simon etal. 2007, Kipping 2009a, 2009b). Sato and Asada (2009) investigated effects of mutual transitsby an extrasolar planet with a companion on light curves. In particular, they studied how theeffects depend on the companion’s orbital velocity. Furthermore, extrasolar mutual transitswere discussed as a complementary method of measuring the system’s parameters such as aplanet-companion’s separation and thereby of identifying them as a true binary, planet-satellitesystem or others.Their method has analogies in classical ones for eclipsing binaries (e.g., Binnendijk 1960,Aitken 1964). A major difference is that occultation of one faint object by the other transiting http://kepler.nasa.gov/ increase in light curves, whereas eclipsing binaries make adecrease. What is more important is that, in both cases where one faint object transits theother and vice versa, changes are made in the light curves due to mutual transits even if nolight emissions come from the faint objects. In a single transit, on the other hand, thermalemissions from a transiting object at lower temperature make a difference in light curves duringthe secondary eclipse, when the object moves behind a parent star as observed for instance forHD 209458b (Deming et al. 2005).This paper is organized as follows. In section 2, we consider effects of the orbital in-clination and eccentricity of a companion on light curves. For simplicity, henceforth, such acompanion orbiting around a host planet is called a ‘moon’ even if it is not a satellite but acomponent of a hypothetical binary planet. In section 3, we present a formulation for param-eter determinations. Some numerical examples are also presented. Section 4 is devoted to theconclusion.
2. Effects of the Orbital Eccentricity of a Transiting ‘Moon’ on Light Curves
The time duration of a transit, say a few hours, is much longer than the orbital periodof an extrasolar planet, say a few days or greater. In one transit, the effect of the motion ofthe moon is much larger than that of the planet orbiting around a star. During the transit,therefore, we employ a constant velocity approximation only for the orbital motion of a planet-moon system around their host star. For a short separation case, on the other hand, we takeaccount of the eccentric orbit of the moon, because the orbital period of such a moon aroundthe planet may be comparable to (or shorter than) the timescale of the transit.The co-planar assumption that the orbital plane of a moon around its primary objectis the same as that of the planet in orbit around the host star seems reasonable because itseems that planets are born from fragmentations of a single proto-stellar disk and thus theirspins and orbital angular momentum are nearly parallel to the spin axis of the disk. Irregularsatellites such as Triton, however, have significant inclinations presumably through captureprocesses. This requires that we should include the effect of orbital inclinations. We assumeonly the moon’s orbital inclination, because it makes substantial effects on mutual transitlight curves. Inclinations of the planet’s orbital plane have been well understood and alreadyobserved (Charbonneau et al. 2000).Here we list our assumptions for clarity. • The inclination angle of the COM of the planet and moon is fixed at 90 degrees. • We take account of the inclination angle of the moon’s orbit. • We assume that the planet-moon COM has a constant velocity (during a transit by theplanet in front of the host star). 3
The longitude of ascending node of the moon equals zero. • The planet-moon COM orbit has zero eccentricity. • We assume no limb darkening effects.For an eccentric orbit, we generally have the Kepler equation as t = t + T π ( u − e sin u ) , (1)where t , T , e and u denote the time of periastron passage, orbital period, eccentricity andeccentric anomaly, respectively (e.g., Danby 1988, Roy 1988, Murray and Dermott 1999). Inthe following, we use the true anomaly f instead of the eccentric anomaly u (e.g., Danby 1988,Roy 1988, Murray and Dermott 1999). They are related bytan f s e − e tan u . (2)The distance between the orbiting body and a focus of the ellipse is written as r = a (1 − e )1 + e cos f . (3)We use these equations for describing a moon orbiting around a planet. We denote themean motion of the moon in orbit around the primary as n m ≡ π/T m , where the subscript m means the moon’s quantity. The subscript p denotes the planet’s quantity.For investigating transits, we need the transverse position x and velocity v of each object.We denote those of the COM for planet-moon systems as x CM and v CM , respectively, where theorigin of x is chosen as the center of the star. We assume v CM as constant during the transit.The position and velocity of each planet with mass M p and M m in the planet-moon system as x i and v i ( i = p, m ), respectively. The direction of the observer’s line of sight is specified by theargument of pericenter as an angle denoted by ω m (See also Fig. 1). We express the transverseposition as x CM = v CM ( t − t CM ) , (4) x p = x CM + a p [(cos u − e m ) cos ω m + q − e m sin u sin ω m ] , (5) x m = x CM − a m [(cos u − e m ) cos ω m + q − e m sin u sin ω m ] , (6)where the semimajor axis of the orbit of each object around their COM is denoted by a i , and t CM means the time when the binary’s common center of mass passes in front of the center ofthe host star. In terms of the true anomaly, they are rewritten as x p = x CM + a p (1 − e m ) cos( f + ω m )1 + e m cos f , (7) x m = x CM − a m (1 − e m ) cos( f + ω m )1 + e m cos f . (8)(See Table 1 for a list of parameters and their definition).The azimuthal velocity of the secondary object around the primary is4 f = r dfdt = a pm n m e m cos f q − e m , (9)where f denotes the true anomaly (Murray and Dermott 2000). Here, a pm and e m denote thesemimajor axis and eccentricity of the moon’s orbit with respect to the host planet. We assumethat both the eccentricity of COM orbit vanishes and the inclination angle of COM equals 90degrees. Hence we can avoid a careful treatment of “sky-projected transverse position”. We denote the intrinsic stellar luminosity as L . The apparent luminosity L ′ due tomutual transits is expressed as L ′ = L × S − ∆ SS , (10)where S = πR s , S p = πR p , S m = πR m , ∆ S = S p + S m − S pm . Here, R s , R p and R m denotethe radii of the host star, planet and moon, and S pm denotes the area of the apparent overlapbetween them, which is seen from the observer. Without loss of generality, we assume that theprimary is larger than the secondary as R p ≥ R m . We investigate light curves by mutual transits due to planet-moon systems. The orbitalvelocity is of the order of a pm n m . Therefore, we have two cases; v CM < a pm n m and v CM > a pm n m .The dimensionless ratio of the moon’s orbital velocity to the planet’s one is defined as W ≡ a pm n m v CM . (11)If v CM < a pm n m , we call it a fast case. If v CM > a pm n m , we call it a slow one. The Earth-Moon ( W = 0 . W = 0 .
8) and Jupiter-Io ( W = 1 .
3) systems representslow, marginal and fast cases, respectively. Figure 2 shows a schematic light curve by mutualtransits.Fig. 3 shows a slow case in circular motion, where we assume R s : R p : R m = 20 : 2 : 1. Weassume also the same mass density for the two transiting objects and hence obtain a p : a m = 1 : 8.Eccentric orbit cases ( W = 6 and e m = 0 .
3) are shown by Figs. 4, 5 and 6 ( ω m = π/ π/
4, respectively). Some parameters are chosen so that effects in the figures can bedistinguished by eye, though such an event is unlikely to be detected by current observations asdiscussed later. For generating the ingress and egress of the various parts of the lightcurve, wedo not use a linear interpolation but compute numerically the apparent overlap area betweenthe objects. Here we assume the same configuration except for the observer’s line of sight. Forsimplicity, we take t = t CM = 0 in these figures.These figures show also the transverse positions of transiting objects with time, which5ould help us to understand the chronological changes in the light curves. In particular, it canbe understood that such characteristic patterns appear only when two objects are in front ofthe star and one of them transits (or occults) the other.
3. Formulation for Parameter Determinations by Transit Method
In all the above cases, the amount of decrease in light curves or the magnitude offluctuations gives the ratios among the radii of the star and two faint objects ( R s , R p , R m ). Thedecrease ratios in the apparent brightness due to transits by the planet and moon are writtenas ∆ p = (cid:18) R p R s (cid:19) , (12)∆ m = (cid:18) R m R s (cid:19) . (13)The stellar radius R s (and mass M S ) are known for instance by its spectral type. Hence, theradii are expressed in terms of observables R s , ∆ p and ∆ m as R p = R s q ∆ p , (14) R m = R s q ∆ m . (15)We define the ratio between the brightness changes by the two objects as∆ ≡ ∆ m ∆ p . (16) Circular Orbit and Orbital Inclination:
First, we discuss a circular orbit in order to simply explain our idea. For more rigorous treatmentof eccentric orbits, please see below, where we will finally give expressions for determining theseparation a pm .Behaviors of apparent light curves depend on W . Therefore, a pm n m (as its ratio to v CM ) can be obtained (Sato and Asada 2009). Seager and Mall´en-Ornelas (2003) presents ananalytic solution of parameter determinations for a single transit in the circular orbit case.Their solution can be used for our case of mutual transits by a planet and moon in front of ahost star. In our case, their equations are rewritten as follows. The duration of the ‘flat part’of the transit ( t F m ) is described bysin (cid:18) t F m πT m (cid:19) = 1 a pm sin I m q ( R p − R m ) − ( a pm cos I m ) , (17)whereas the total transit duration ( t T m ) is done bysin (cid:18) t T m πT m (cid:19) = 1 a pm sin I m q ( R p + R m ) − ( a pm cos I m ) , (18)6here I m denotes the orbital inclination angle of the moon. By combining these equations, theimpact parameter ( b pm ) can be derived as b pm ≡ a pm R p cos I m = vuut (1 − √ ∆) − [sin ( t F m π/T m ) / sin ( t T m π/T m )](1 + √ ∆) − [sin ( t F m π/T m ) / sin ( t T m π/T m )] . (19)The ratio a pm /R p can be derived directly from Eq. (18) as a pm R p = vuut (1 + √ ∆) − b pm [1 − sin ( t T m π/T m )]sin ( t T m π/T m ) . (20)Therefore, one can obtain a pm as R s × ( R p /R s ) × ( a pm /R p ).With a pm and n m in hand, one can thus estimate the total mass of the binary by GM tot = n m a pm from Kepler’s third law, where G denotes the gravitational constant. Theorbital velocity a pm n m gives the mutual force between the binary.If we assume also that the mass density is common for two objects constituting thebinary (this may be reasonable especially for similar size objects as R p ∼ R m ), each mass isdetermined as M p = R p ( R p + R m ) − M tot and M m = R m ( R p + R m ) − M tot , respectively. Therefore,the orbital radius of each body around the COM is obtained as a p = R m ( R p + R m ) − a pm and a m = R p ( R p + R m ) − a pm , respectively. At this point, importantly, the two objects can beidentified as a true binary ( a p > R p ) or planet-satellite system ( a p < R p ). However, a gaseousgiant planet with a rocky satellite would exhibit largely different densities and this may be oneof the most likely scenarios.In a slow spin case, on the other hand, the apparent separation a ⊥ (normal to our lineof sight) is determined as a ⊥ = T v CM from measuring the time lag T between the first andsecond transits because v CM is known above (Sato and Asada 2009). Eccentric Orbit and Edge-on Case:
Henceforth, we take account of the orbital eccentricity of a moon for an edge-on case. Inthis case, intervals between neighboring “hills” are not constant because of the eccentricity.However, a time duration between three successive “hills” is nothing but the orbital period ofthe moon. Therefore, one can measure the period T m . We obtain n m as n m = 2 πT m . (21)A key idea for determining the eccentricity is as follows. As for timescales, we havetwo observable ratios as T /T and T /T . The former is the ratio between the widths ofneighboring hills, whereas the latter is that between the transit intervals. On the other hand,we have two additional parameters e m and ω m to be determined. Importantly, the numberof measurable ratios is the same as that of the parameters that we wish to determine. Inprinciple, therefore, the above two ratios may allow us to determine the two parameters e m and7 m , separately. This will be discussed in detail below.To be more precise, the full width of a “hill” at top and bottom are expressed as (Seealso Figure 2) T top = 2( R p − R m ) V f , (22) T bottom = 2( R p + R m ) V f . (23)Only for symmetric binaries ( R p = R m ), we have T top = 0 and thus true spikes. Otherwise,truncated spikes (or “hills”) appear.For the primary transit, where the moon moves in front of the planet, we have f = π/ − ω m . From Eqs. (9), (22) and (23), therefore, we obtain T top = 2( R p − R m ) a pm n m q − e m e m sin ω m , (24) T bottom = 2( R p + R m ) a pm n m q − e m e m sin ω m . (25)For the circular orbit e m = 0, the second factors in the R.H.S. of these expressions become theunity and the first factors recover the case for e m = 0 (See Sato and Asada 2009).For the secondary transit, where the moon moves behind the planet, we have f = 3 π/ − ω m . From Eqs. (9), (22) and (23), therefore, we obtain T top = 2( R p − R m ) a pm n m q − e m − e m sin ω m , (26) T bottom = 2( R p + R m ) a pm n m q − e m − e m sin ω m . (27)We immediately obtain from Eqs. (24), (25), (26) and (27), T r ≡ T top T top = T bottom T bottom = 1 + e m sin ω m − e m sin ω m . (28)Equation (28) is rewritten as e m sin ω m = T r − T r + 1 ≡ T R , (29)where the R.H.S. can be determined by observations alone. Once either e m or ω m is known,Eq. (29) determines the other.Next, we consider the time interval between the primary and secondary transits. In8rder to compute such an interval, one can use the Kepler’s second law (the constant arealvelocity). After lengthy but straightforward calculations, the area swept from the primarytransit ( f = π/ − ω m ) till the secondary ( f = 3 π/ − ω m ) becomes S = a m b m × (cid:20) π H + 12 sin(2 arcsin H ) (cid:21) , (30)where b m denotes the semiminor axis of the moon’s elliptic orbit and H is defined as H = e m vuut cot ω m − e m + cot ω m . (31)By using Eq. (29) in Eq. (31) for eliminating cot ω m , we obtain H = vuut e m − T R − T R . (32)It is convenient to use T /T instead of T /T , because T /T gives a simpler expressionthan T /T , which is used in the above explanation of the key idea. The total area is S = πa m b m . Therefore, we find T T = S S = 12 + 1 π arcsin H + 1 π H s − e m − T R . (33)This includes e m without ω m . Hence by using this relation, one can determine separately theeccentricity by measuring the time intervals. We should note that Eq. (33) is valid even for ageneral case with a certain inclination angle.Here, we consider a small eccentricity approximation, which may be useful for a quickerestimation of the parameters. For small e m , Eq. (31) is expanded as H = e m cos ω m + O ( e m ) . (34)Substitution of this into Eq. (33) gives us T T m = 12 + 2 e m π cos ω m + O ( e m ) . (35)The correction to the circular case ( T /T m = 1 /
2) is 2 e m cos ω m /π ∼ . e m cos ω m . Even for ω m ∼ e m = 0 . T and T as δT ≡ T − T . (36)Replacement ω m by ω m + π changes Eq. (35) into T T m = 12 − e m π cos ω m + O ( e m ) . (37)By using Eqs. (35) and (37), we obtain 9 m cos ω m = πδT T m + O ( e ) . (38)Let us consider every cases separately.(I) If and only if the L.H.S. of Eqs. (29) and (38) vanish, e m does. This means a circularmotion and thus the observer’s direction ω m becomes meaningless.(II) For a case when the L.H.S. of Eq. (29) vanishes but that of Eq. (38) does not, wefind e m = 0 and sin ω m = 0, namely ω m = 0 (mod π ) . (39)Eq. (38) immediately gives e m = πδT T m . (40)Hence, the orbital eccentricity is determined.(III) If the L.H.S. in Eq. (38) vanishes but that in (29) does not, we find e m = 0 andcos ω m = 0, namely ω m = π π ) . (41)Eq. (29) immediately gives e m = T R . (42)Hence, the orbital eccentricity is measured.(VI) A general case in which the L.H.S. of neither Eqs. (29) nor (38) vanish: By dividingEq. (38) by Eq. (29), we obtaincot ω m = π δTT m T R , (43)where the R.H.S. can be determined by observations alone and hence this equation gives us theobserver’s direction ω m . By substituting the determined ω m into Eq. (29), one can find thevalue of the eccentricity.Up to this point, e m and ω m both are determined. Eqs. (24) and (25) are rewritten as a pm = T m ( R p − R m ) πT top q − e m e m sin ω m = T m ( R p − R m )( T top + T top ) q − e m πT top T top = ( R p − R m )2 π T m T top + T m T top ! q − e m , (44) a pm = T m ( R p + R m ) πT bottom q − e m e m sin ω m = T m ( R p + R m )( T bottom + T bottom ) q − e m πT bottom T bottom
10 ( R p + R m )2 π (cid:18) T m T bottom + T m T bottom (cid:19) q − e m , (45)respectively, where we used Eq. (29) If and only if R p = R m , we obtain T top = T top = 0 andEq. (44) thus becomes undetermined, whereas Eq. (45) is still well-defined.When one wishes to consider the secondary transit instead of the first, one can use a pm = T m ( R p − R m ) πT top q − e m − e m sin ω m , (46) a pm = T m ( R p + R m ) πT bottom q − e m − e m sin ω m . (47)They are obtained by replacing ω m with ω m + π in Eqs. (44) and (45). By noting T top (1 + T R ) = T bottom (1 − T R ) and T top (1 + T R ) = T bottom (1 − T R ), one can show that Eqs. (46) and (47) agreewith Eqs. (44) and (45), respectively. By using one of these expressions, we can thus measurethe semimajor axis of the eccentric orbit. In terms of the decrease in apparent brightness, Eqs.(44) and (45) are written as a pm R s = ( q ∆ p − √ ∆ m )2 π T m T top + T m T top ! q − e m , (48) a pm R s = ( q ∆ p + √ ∆ m )2 π (cid:18) T m T bottom + T m T bottom (cid:19) q − e m , (49)where we used Eqs. (14) and (15).Determination of the semimajor axis is sensitive to measurement errors in the widthsof the hills. This statement can be proven by using Eq. (49). For simplicity, we assume∆ p ∼ ∆ m ∼ ∆ and T bottom ∼ T bottom ∼ T bottom , so that Eq. (49) can be reduced to a pm ∼ π R s √ ∆ T m T bottom q − e m . (50)The logarithmic derivative of this becomes da pm a pm ∼ dR s R s + d ∆2∆ + dT m T m − dT bottom T bottom − e m de m (1 − e m ) . (51)We focus on dT m /T m and dT bottom /T bottom , because we can expect much more accurate mea-surements for R s and ∆, and the last term involving e m may be relatively small ( e m < de m < dT m ∼ dT bottom , while T m ≫ T bottom . Therefore, dT m /T m ≪ dT bottom /T bottom , which means that accurate measurements of T bottom are crucial forthe determination of a pm .Figure 7 shows a flow chart of the parameter determinations that are discussed above.The above formulation for parameter determinations actually recovers the correct values inFigs. 4-6. In the numerical examples, the original parameters are retrieved within twentypercents. Eccentric Orbit and Orbital Inclination: I m = 90 deg.) and an inclinationcase ( I m = 88 deg.). Let z -axis denote the axis normal to the x -axis on the celestial sphere. Wedefine the distance of a ‘moon’ from the z -axis at the initial time of the mutual transit as s b = vuut ( R p + R m ) − a pm (1 − e m )1 + e m cos f ! cos I m , (52)where the subscript b means that the quantity is related with T bottom and T bottom as shownbelow (See also Fig. 9). Similarly, when the ‘flat part’ of the spike in light curves starts (orends), we define the distance of a moon from the z -axis at this epoch as s t = vuut ( R p − R m ) − a pm (1 − e m )1 + e m cos f ! cos I m , (53)where the subscript t means that the quantity is related with T top and T top as shown below.Therefore, we obtain the duration T top as T top = 2 s t V f , (54)where V f is given by Eq. (9). For the primary transit ( f = π/ − ω m ), we thus obtain T top = 2 s t q − e m a pm n m (1 + e m sin ω m ) , (55)whereas for the secondary ( f = 3 π/ − ω m ), we have T top = 2 s t q − e m a pm n m (1 − e m sin ω m ) . (56)Here, we define s t and s t as s t = vuut ( R p − R m ) − a pm (1 − e m )1 + e m sin ω m ! cos I m , (57) s t = vuut ( R p − R m ) − a pm (1 − e m )1 − e m sin ω m ! cos I m . (58)In the similar manner, we obtain the width of the spikes at the bottom as T bottom = 2 s b V f . (59)For the primary transit ( f = π/ − ω m ), we thus obtain T bottom = 2 s b q − e m a pm n m (1 + e m sin ω m ) , (60)whereas for the secondary ( f = 3 π/ − ω m ), we have T bottom = 2 s b q − e m a pm n m (1 − e m sin ω m ) , (61)where we define s b and s b as 12 b = vuut ( R p + R m ) − a pm (1 − e m )1 + e m sin ω m ! cos I m , (62) s b = vuut ( R p + R m ) − a pm (1 − e m )1 − e m sin ω m ! cos I m . (63)Because of the orbital inclination, we have to consider T top and T bottom , separately. Wedefine the ratios as T rtop ≡ T top T top , (64)and T rbottom ≡ T bottom T bottom . (65)Substitutions of Eqs. (55), (56), (60), (61) into these ratios lead to T rtop = s t s t e m sin ω m − e m sin ω m , (66) T rbottom = s b s b e m sin ω m − e m sin ω m . (67)For the edge-on case ( I m = 90 deg.), we obtain s t /s t = s b /s b = 1. Then, we have T rtop = T rbottom . For a general case ( I m = 90 deg.), on the other hand, we find T rtop = T rbottom .We thus expect that a ratio between them will give us the information about the orbitalinclination. The ratio is T rtop T rbottom = s t s t s b s b . (68)The L.H.S. can be measured by observations.There are four unknown quantities a pm , e m , I m and ω m . We have four equations of (66),(67), (68) and the last one that can be chosen out of (55), (56), (60) and (61). Therefore, onecan determine the quantities a pm , e m , I m and ω m by using these equations for observations.For practical observations, data fittings at the slope of light curves are used, instead of tran-sit durations, for determinations of the orbital inclination angle (Charbonneau et al. 2000).Nevertheless, an analytic solution is necessarily worthwhile to understand the properties of agiven physical system, even when numerical fits are in practice the best way to determine thesystem parameters.A partial transit occurs if the apparent impact parameter of the moon is in ( R p − R m ,R p + R m ). For the primary transit ( f = π/ − ω m ), it occurs if the orbital inclination angle satisfies R p − R m a pm e m sin ω m − e m < cos I m < R p + R m a pm e m sin ω m − e m . (69)For the secondary one ( f = 3 π/ − ω m ), the condition of a partial transit becomes R p − R m a pm − e m sin ω m − e m < cos I m < R p + R m a pm − e m sin ω m − e m . (70)13uch a partial transit by a moon orbiting a host planet produces a ‘U’-shaped spike in lightcurves (See Fig. 10). We have presented a formalism for parameter determinations. Before closing this section,let us make brief comments on typical timescales and amplitudes in the brightness changes.The timescale of a brightness change due to a giant planet is about R p a pm n m ∼ × R p × km 10km/s a pm n m ! sec. (71)Therefore, detections of such fluctuations due to mutual transits of extrasolar planet-moonsystems require frequent observations, say every hour. Furthermore, higher frequency (e.g.,every ten minutes) is necessary for parameter estimations of the system.Let us mention a connection of the present result with space telescopes in operation.Decrease in apparent luminosity due to the secondary planet is O ( R m /R s ). Besides the timeresolution (or observation frequency) and mission lifetimes, detection limits by COROT withthe achieved accuracy of photometric measurements (700 ppm in one hour) could put R m /R s ∼ × − . The nominal integration time is 32 sec. but co-added over 8.5 min. except for1000 selected targets for which the nominal sampling is preserved. By the Kepler missionwith expected 20 ppm differential sensitivity for solar-like stars with m V = 12, the lower limitwill be reduced to R m /R s ∼ × − . An analogy of the Earth-Moon ( R m /R s ∼ . × − , W ∼ .
03) and Jupiter-Ganymede ( R m /R s ∼ × − , W ∼ .
8) will be marginally detectable.Observations both with high frequency (at least during the time of transits) and with goodphotometric sensitivity are desired for future detections of mutual transits.
For Roche limit, we have a pm < βR H , (72)where β denotes a numerical coefficient 0 < β < R H is Hill radius (See Domingos etal. 2006 for more detailed stability arguments by numerical computations). For simplicity, weassume M s ≫ M p ≫ M m . Then, we have a pm ≈ a m and R H is approximated as R H = (cid:18) M p M s (cid:19) / d p , (73)where M s denotes a host star mass and d p denotes the orbital radius of a planet orbiting thestar. Kepler’s third law gives a pm n m and v CM as a pm n m = s GM p a m , (74) v CM = s GM s d p . (75)14ombining these relations, therefore, we find W > / β − / (cid:18) M p M s (cid:19) / . (76)If one assumes M p ∼ M J (Jupiter mass), we obtain W > − . This lower bound is less severe.On the other hand, there is a stringent constraint that the moon’s closest approach r min cannotbe within the planetary radius. We thus have a m > r min − e m . (77)This leads to W < s M p M s d p r min (1 − e m ) . (78)If one assumes the jovian mass and radius ( R J ), this is rewritten as W < . d p ! / (cid:18) M ⊙ M s (cid:19) / M J M p ! / (cid:18) R J r min (cid:19) / (1 − e m ) / . (79)For W = 6 and e m = 0 .
3, we obtain d p >
25 AU. Namely, a planet with a long orbital period T p >
125 years is required. Kepler mission for several years is unlikely to see a transit by sucha long period planet.Next, we consider a constraint that a mutual transit can occur. The transit duration fora planet in circular orbit is D ∼ T p R s πd p , (80)where we assume the maximum duration by taking the vanishing impact parameter (See Seagerand Mall´en-Ornelas for a more accurate form). As this limiting case for the Roche limit, weobtain the fastest case of a ‘moon’ as a m ∼ β (cid:18) M p M s (cid:19) / d p . (81)Using the Kepler’s third law, this leads to T m ∼ β ! / T p . (82)For our fast case, we require T m < D , which gives a bound on β as β < / R s πd p ! / . (83)We substitute this into β of Eq. (76) so that we can obtain W > πd p R s ! / (cid:18) M p M s (cid:19) / ∼ . d p d J ! / (cid:18) R ⊙ R s (cid:19) / (cid:18) M p M J (cid:19) / (cid:18) M ⊙ M s (cid:19) / . T p ! / (cid:18) R ⊙ R s (cid:19) / (cid:18) M p M J (cid:19) / (cid:18) M ⊙ M s (cid:19) / , (84)where d J denotes the mean orbital radius of the Jupiter.On the other hand, the dynamical arguments put an upper bound by Eq. (79). This isrewritten as W < . T p ! / (cid:18) M ⊙ M s (cid:19) / M J M p ! / (cid:18) R J r min (cid:19) / (1 − e m ) / . (85)Therefore, there exists a narrow band as shown by Fig. 11. Outside this range, the proposedmethod cannot work.
4. Conclusion
We have shown that light curves by mutual transits of an extrasolar planet with a‘moon’ depend on the moon’s orbital eccentricity, especially for small separation (fast) cases, inwhich occultation of one faint object by the other transiting a parent star causes an apparentincrease in light curves and such characteristic fluctuations with the same height repeatedlyappear. We have also presented a formulation for determining the parameters such as theorbital eccentricity, inclination, semimajor axis and the direction of the observer’s line of sight.This will be useful for probing the nature of the transiting planet-moon system.When actual light curves are analyzed, we should incorporate (1) photometric correctionssuch as limb darkenings, and (2) perturbations as three (or more)-body gravitating interactions(e.g., Danby 1988, Murray and Dermott 2000).We would like to thank the referee for invaluable comments on the manuscript. Wewould like to thank S. Ida, S. Inutsuka. E. Kokubo and Y. Suto for stimulating conversationsand encouragements. This work was supported in part (H.A) by a Japanese Grant-in-Aid forScientific Research from the Ministry of Education, No. 21540252.
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Symbol Definition T m Orbital period of an extrasolar ‘moon’ around a planet n m Mean motion of a moon (= 2 π/T m ) a m Semimajor axis of a moon’s orbit w.r.t. a planet-moon’s center of mass a p Semimajor axis of a planet’s orbit w.r.t. a planet-moon’s center of mass a pm Semimajor axis of a moon’s orbit around a planet (= a p + a m ) a ⊥ Apparent separation of a planet-moon system R s Radius of a host star R p Radius of a planet R m Radius of a moon M p Mass of a planet M m Mass of a moon M tot M p + M m x CM Transverse position of a planet-moon’s center of mass v CM Transverse velocity of a planet-moon’s center of mass x p Transverse position of a planet x m Transverse position of a moon e Orbital eccentricity of the moon t Time of periastron passage of the moon ω m Argument of pericenter of the moon t CM Time when the planet-moon’s center of mass passes across the star’s center∆ p Decrease rate in apparent brightness due to the planet transit∆ m Decrease rate in apparent brightness due to the moon transit T top Time duration: width of a hill’s top at the primary transit in light curves T bottom Time duration: width of a hill’s bottom at the primary transit in light curves T top Time duration: width of a hill’s top at the secondary transit in light curves T bottom Time duration: width of a hill’s bottom at the secondary transit in light curves T Time lag from the first transit to the secondary T Time lag from the secondary transit to the first δT T − T ig. 1. Direction of the line of sight. It is denoted as ω m , which is the argument of pericenter. ig. 2. Schematic figure of a light curve due to a mutually transiting planet and moon in front of theirhost star. ig. 3. Light curves: Solid red one denotes the zero limit of a ‘moon’ orbital motion as a reference( W ≡ a pm n m /v CM = 0). Dashed green one is a marginal spin case (large separation) for W = 1 (Satoand Asada, 2009). The vertical axis denotes the apparent luminosity (in percents). The horizontal oneis time in units of the half crossing time of the star by the COM of the binary, defined as R s /v CM . Ifone takes M s = M ⊙ , R s = R ⊙ and v CM = 30 km/s (namely, 1 AU distance from the host star), t = 1corresponds to ≈ R s : R p : R m = 20 : 2 : 1, and a pm /R s = 0 . % tW=6, e_m=0.3, omega=90 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=90 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=90 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=90 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=90 deg. Fig. 4.
Top panel: a light curve for a fast case (small separation) as W = 6 and e m = 0 .
3. The radius andseparation are the same as those in Fig. 3. The observer’s direction is ω m = π/
2. Brightness fluctuationsappear with T top /T m = 0 . T bottom /T m = 0 . T top /T m = 0 . T bottom /T m = 0 . T /T m = 0 . T /T m = 0 .
49 (normalized by T m , the orbital period of the moon: T m = T + T ). We obtain T r ∼ . e m sin ω m ∼ .
3, whereas Eq. (38) approximately gives us e m cos ω m ∼
0. Therefore,we recover well the parameters as e m ∼ . ω m ∼ π/
2, even in the linear approximation in e m . Finally,Eq. (49) tells a pm /R s ∼ .
9. Bottom panel: motion of each body in the direction of x normalized by R s (solid red for the primary and dotted green for the secondary). When one faint object transits or occultsthe other in front of the host star, mutual transits occur and a ‘hill’ appears in the light curve. % tW=6, e_m=0.3, omega=0 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=0 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=0 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=0 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=0 deg. Fig. 5.
Top panel: a light curve for all the parameters same as those in Fig. 4, except for the ob-server’s direction as ω m = 0. Brightness fluctuations appear with T top /T m = 0 . T bottom /T m = 0 . T top /T m = 0 . T bottom /T m = 0 . T /T m = 0 .
69 and T /T m = 0 .
31. We obtain T r ∼ . e m sin ω m ∼
0, whereas Eq. (38) approximately gives us e m cos ω m ∼ .
3. Therefore, werecover well the parameters as e m ∼ . ω m ∼
0, even in the linear approximation in e m . Finally, Eq.(49) tells a pm /R s ∼ .
9. Bottom panel: motion of each body (solid red for the primary and dotted greenfor the secondary). % tW=6, e_p=0.3, omega=45 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=45 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=45 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=45 deg.-1-0.5 0 0.5 1-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x tW=6, e_m=0.3, omega=45 deg. Fig. 6.
Top panel: a light curve for all the parameters same as those in Fig. 4, except for the ob-server’s direction as ω m = π/
4. Brightness fluctuations appear with T top /T m = 0 . T bottom /T m = 0 . T top /T m = 0 . T bottom /T m = 0 . T /T m = 0 .
62 and T /T m = 0 .
38. We obtain T r ∼ . e m sin ω m ∼ .
2, whereas Eq. (38) approximately gives us e m cos ω m ∼ .
2. Therefore, werecover well the parameters as e m ∼ . ω m ∼ π/
4, even in the linear approximation in e m . Finally,Eq. (49) tells a pm /R s ∼ .
9. Bottom panel: motion of each body (solid red for the primary and dottedgreen for the secondary). ig. 7. Flow chart of parameter determinations. Starting from measurements of brightness changes, thesemimajor axis a pm can be finally determined for a fast case. % t 98.6 98.7 98.8 98.9 99 99.1 99.2 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 % t Fig. 8.
Difference in light curves due to orbital inclinations of a ‘moon’. All parameters but for theinclination angle are the same as those in Fig. 4. The solid (red) curve denotes I m = 90 degree case, whilethe dashed (green) one means a case of I m = 88 deg. Because of the orbital inclination, the duration of amutual transit by the satellite is shortened. ig. 9. Definition of s b and s t : The total transit duration T bottom is given by s b , whereas the ‘flat part’of the hill T top is done by s t . % t 98.6 98.7 98.8 98.9 99 99.1 99.2 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 % t Fig. 10.
Light curve by a partial mutual transit of a satellite. All parameters but for the inclinationangle are the same as those in Fig. 4. The solid (red) curve denotes I m = 90 degree case, while the dashed(green) one means a case of I m = 86 degree for a partial transit. Such a partial transit case produces a‘U’-shaped hill in light curves. W T_p [year] 0 2 4 6 8 10 12 0.01 0.1 1 10 100 W T_p [year]
Fig. 11.
Possible bound on W and T p . Here, we assume M s = M ⊙ , R s = R ⊙ , M p = M J , r min = R J and e m = 0. The shaded (green) region denotes prohibited regions of the parameters that are constrained byEqs. (84) and (85).= 0. The shaded (green) region denotes prohibited regions of the parameters that are constrained byEqs. (84) and (85).