Early UV Ingress in WASP-12B: Measuring Planetary Magnetic Fields 3

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AL VIGESI AP1B ESRN LNTR MAGNETIC PLANETARY MEASURING WASP-12B: IN INGRESS UV EARLY (4 colo hsc n srnm,Uiest fS nrw,N Andrews, St of University Astronomy, and Physics of School colo hsc n srnm,Uiest fS nrw,N Andrews, St of University Astronomy, and Physics of School colo hsc n srnm,Uiest fS nrw,N Andrews, St of University Astronomy, and Physics of School − M 5) 1. J M × A INTRODUCTION and p T 10 E tl mltajv 04/20/08 v. emulateapj style X 1 = 6 lntsa neatos—paesadstlie:idvda (WASP- individual satellites: and planets — interactions planet-star .I h ofie ae h hc lasfrsdrcl ha ft of ahead directly forms always shock the case, confined the In K. R n aelts antcfils—sas ooa tr:idvda ( individual stars: — coronae outflows stars: winds, — fields stars: magnetic satellites: and J . 41 r h asadradius and mass the are B M p J ntecs fWS-2,w eiea pe ii fabout of limit upper an derive we WASP-12b, of case the In . n radius and rf eso etme 9 2018 19, September version Draft R .A Vidotto A. A. p ABSTRACT h Helling Ch. .Jardine M. = and and tr ihmass with velocity star, ambient azimuthal the account. relative and into taken motion the formed is orbital medium be if planetary we actually the planet expected Here, can between shock is the zone. orbital bow interaction shock around WASP-12b a planet-wind that bow at the demonstrate no in subsonic form that still to is concluding a wind found distance, using They the WASP-12b 1958). of (Parker that model distance wind the driven derived at mass-loss thermally LHV10 the velocity wind properties, wind of solar wind the cause solar typical shock adopting the a and as Assuming planet’s bow rate the wind ingress. a of stellar interaction early a the of with to due formation magnetosphere planet ingress. the the the early around of an investigated egress of instead late LHV10 curve, a radiation- light produce a transit would planetary that tail 209458b al. be cometary Heuvel HD et driven also den Ehrenreich for van However, could demonstrated & tails. (2008) Helling, Asymmetries cometary (Lai, by as LHV10). Earth produced system hereafter, the planet-star 2010, transiting from the seen of appearance the ta ria aisof radius orbital an at nobtlpro of period orbital an ieto,a h lntmvsa elra ria ve- orbital Keplerian a im- azimuthal at of the particles moves locity from planet coronal mainly the of comes as flux planet direction, the the on star, pacting the to proximity AP1bobt t otsa alt- main-sequence late-F (a star host its orbits WASP-12b rhHuh tAdes Y69S UK 9SS, KY16 Andrews, St Haugh, orth rhHuh tAdes Y69S UK 9SS, KY16 Andrews, St Haugh, orth rhHuh tAdes Y69S UK 9SS, KY16 Andrews, St Haugh, orth u K ( = hwda al nrs compared ingress early an showed b oli etn iiso planetary on limits setting in tool l eeryU nrs slkl ob a be to likely is ingress UV early le niin htmgtla othe to lead might that onditions scue yabwsokahead shock bow a by caused is M neo h lsatemperature. plasma the on ence GM etr ewe h lntand planet the between centers tad()teselrmagnetic stellar the (2) and it tla antcfil sstrong is field magnetic stellar ae o lsatemperatures plasma for lanet ∗ 1 = P ∗ /R orb R . 35 orb e,asokcpbeof capable shock a ses, orb 1 = M ) 1 0 = . / ⊙ 9d(0) u oisclose its to Due (H09). d 09 2 n radius and ∼ epae,btin but planet, he . 2 U=3 = AU 023 2)—planets — 12b) 3 ms km 230 AP1)— WASP-12) FIELDS B p 4G. 24 = R − ∗ 1 . 15 1 = rudthe around R . 57 ∗ with , R ⊙ ) 2 Vidotto, Jardine & Helling star. Therefore, stellar coronal material is compressed tance, the stellar wind is still accelerating and subsonic ahead of the planetary orbital motion, possibly forming (ur < cs). In this case, conditions for an ahead-shock a bow shock ahead of the planet. If such compressed ma- will more probably be met. terial is optically thin, the planetary transit light curve Here, we adopt the estimates of LHV10 to derive the is symmetrical with respect to the ingress and the egress minimum density of the stellar plasma at the orbital ra- of the planet. Indeed, this is the case when the transit is dius of WASP-12b. From the observations presented observed at optical wavelengths (H09). However, if the in Fossati et al. (2010a), LHV10 estimated the column shocked material ahead of the planet can absorb enough density of the absorbing gas around the planet to be stellar radiation, the observer will note an early ingress & 1.4 × 1013 cm−2. For that, they assumed τ = 1 in the of the planet in the stellar disk, but no difference will absorption lines of Mg. Because the time resolution of be seen at the time of egress, as the shocked material is the transit measurement is sparse, it is difficult to place a present only ahead of the planetary motion. firm constrain on the thickness of the absorbing material, Here, we investigate under which conditions the inter- which could either extend all the way to the planetary action of a planet with the stellar coronal plasma could surface or be limited to a geometrically thin shocked re- lead to the formation of a bow shock ahead of the plan- gion. Through time differences of the optical/near-UV etary orbital motion, and therefore, explain the early ingresses, LHV10 estimated that the stand-off distance ingress observable in the near-UV. Our shock model is between the shock and the center of the planet is about −3 described in Section 2. Although we know the orbital 4.2 Rp, leading to a Mg density nMg & 400 cm . Here, radius of the planet, we do not know if at this radius the we convert this minimum density to hydrogen number stellar magnetic field is still capable of confining the hot density by using the observed metallicity of WASP-12, gas of the stellar corona, or if this gas is escaping in a [M/H]=0.3 (H09) wind. Therefore, we investigate the validity of our model n with respect to different stellar coronal conditions in Sec- Mg = 10(ǫMg−ǫH )100.3 ≃ 6.76 × 10−5, (2) tions 3 and 4. Conclusions are presented in Section 5. nH 2. THE SHOCK MODEL where ǫMg = 7.53 and ǫH = 12.00 are the solar element abundances for Mg and H, respectively (Grevesse et al. A bow shock around a planet is formed when the 2007). Therefore, the minimum density of the shocked relative motion between the planet and the stellar stellar coronal plasma is corona/wind is supersonic. The shock configuration depends on the direction of the flux of particles that ar- 400 cm−3 n ≃ ≃ 6 × 106 cm−3. (3) rives at the planet. We illustrate two different limits H 6.76 × 10−5 of the shock configuration and an intermediate case in Figure 1, where θ is the deflection angle between the az- The density behind such a shock would, in the adiabatic imuthal direction of the planetary motion and n, a vector limit, be at most four times the density ahead of the that defines the outward direction of the shock. As seen shock. From Equation (3), this would then require a from the planet, −n is the velocity of the impacting ma- pre-shock coronal density of terial. − n ≃ 1.5 × 106 cm 3 (4) The first shock limit, a “dayside-shock”, occurs when obs the dominant flux of particles impacting on the planet in the external ambient medium at the orbital radius of arises from the (radial) wind of its host star. For in- the planet. stance, the impact of the supersonic solar wind forms a The temperature of the absorbing material must be bow shock at the dayside of Earth’s magnetosphere (i.e., such as to allow for the presence of Mg II lines. For at the side that faces the Sun). This condition is illus- an adiabatic shock, the temperature immediately behind trated in Figure 1(a) and is met when ur > cs, where the shock depends on the squared Mach number of the ur and cs are the local radial stellar wind velocity and flow impacting on the planet. To determine the temper- sound speed, respectively. ature and ionization structures of the atmosphere of the A second shock limit, an “ahead-shock”, occurs when planet, and therefore quantitatively evaluate the optical the dominant flux of particles impacting on the planet depth of Mg II lines, detailed radiative transfer models arises from the relative azimuthal velocity between the are needed. planetary orbital motion and the ambient plasma. This condition is especially important when the planet orbits 3. HYDROSTATIC CORONA at a close distance to the star, and therefore, possesses a In this section, we assume that the orbit of the planet high uK. In this case, the velocity of the particles that lies within the stellar magnetosphere. In this case, the the planet ‘sees’ is supersonic if ∆u = |uK − uϕ| > cs, planet moves through a medium that is confined by a where uϕ is the azimuthal velocity of the stellar corona. rigid stellar magnetic field and so rotates at the stellar This condition is illustrated in Figure 1(b). rotation rate. The relative azimuthal velocity between For intermediate cases, both the wind and the az- the planet and this medium is therefore imuthal relative velocities will contribute to the forma- 1/2 tion of a shock around the planet (Figure 1(c)) and the GM∗ 2πRorb deflection angle θ is given by ∆u = |uK − uϕ,cor| = − , (5) Rorb P∗

ur θ = atan . (1) where uϕ,cor =2πRorb/P∗ is the velocity of the medium |uK − uϕ| corotating with the star, and P∗ is the stellar period of In general, for a planet orbiting its host star at a close dis- rotation. If ∆u>cs, a bow shock forms around the Early UV Ingress in WASP-12b: Measuring Planetary Magnetic Fields 3

Fig. 1.— Sketch of shock types (not to scale): (a) dayside-shock (θ = 90o), (b) ahead-shock (θ = 0o), and (c) intermediate case. Arrows radially leaving the star depict the stellar wind, dashed semi-circles represent the orbital path, θ is the deﬂection angle between n =∆u−ur and the relative azimuthal velocity of the planet ∆u. planet. We may therefore deﬁne a critical stellar rotation 30 period such that ∆u = cs and 25 No shock 2πRorb 20 |Pcrit,∗| = . (6) |uK − cs| 15 ∗ Figure 2 presents Pcrit, as a function of the coronal 10 temperature T . The shaded areas show regions on the Pcrit,∗ − T parameter space where no shock is formed. In 5

∗ (d) our notation, Pcrit, < 0 refers to cases where the star is * 0 K (WASP-12b) crit counterrotating with respect to the planetary orbital ro- 6 tation (e.g., WASP-8b; Queloz et al. 2010). The vertical P -5 line in Figure 2 represents a critical coronal temperature -10 where uK = cs and |Pcrit,∗| → ∞ (cf. Equation (6)). For = 4.16 x 10 -15 crit WASP-12, we obtain T No shock -20 GM∗m 6 (M∗/M⊙)µ 6 Tcrit = = 23.1×10 K=4.16×10 K, -25 RorbkB (Rorb/R⊙) (7) -30 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 where m = µmp is the mean particle mass, kB is the T (106 K) Boltzmann constant, and mp is the proton mass. We adopt µ = 0.66. Above this temperature, no shock for- Fig. 2.— Critical stellar rotation period (Equation (6)) required for the formation of a bow shock as a function of the environment mation is possible at any positive P∗. For all the re- 1 temperature. Negative periods refer to cases where the planet is in maining areas of the plot an ahead-shock will develop. a retrograde orbit. Although P∗ is unknown, WASP-12 is likely to be a slow rotator (Fossati et al. 2010b). With this information, the density was scaled to match the observationally derived 6 −3 presence of an ahead-shock for WASP-12b constrains the value of nobs ≃ 1.5 × 10 cm (Equation (4)), which is coronal temperature to be T . (4 − 5) × 106 K. the minimum density required for the detection of the For an isothermal corona, the density is early ingress. Except for the very low temperature case (black solid line), a typical solar coronal base density GM∗/R∗ R∗ 8 −3 n(Rorb)= n0 exp − 1 (i.e., n0 ∼ 10 cm ; Withbroe 1988) provides the con- kBT/m Rorb dition for the existence of a shock. 2π2R2/P 2 R2 We note that the observationally derived stand-off dis- + ∗ ∗ orb − 1 , (8) k T/m R2 tance from the shock to the center of the planet can B ∗ be taken as approximately the extent of the planetary where n0 is the density at the coronal base. From Equa- magnetosphere rM ≃ 4.2 Rp (LHV10). Pressure balance tion (8), we note that the more slowly the star rotates, between the coronal total pressure and the planet total the lower is the density at any given radius. For the pressure requires that, at rM , very slow stellar rotation rates expected for stars with 2 2 2 ρc∆u Bc(Rorb) Bp(rM ) detected transiting planets, P∗ has little influence on the + + pc = + pp, (9) density. The solid lines in Figure 3 show the density 2 4π 4π variation for a range of orbital radius for three differ- where ρc = mnobs, pc and Bc(Rorb) are the local coronal ent temperatures and a solar value P∗ = 26 d. The mass density, thermal pressure, and magnetic field inten- 1 sity, and pp and Bp(rM ) are the planet thermal pressure A “behind-shock” is formed when uK − uϕ,cor < 0, or ≃ and magnetic field intensity at rM . Neglecting the ki- Pcrit,∗ < Porb 1 d for WASP-12b. In such situation, the planet 2 orbits beyond the Keplerian corotation radius. The coronal plasma netic term and the thermal pressures , we may therefore lags behind the planetary motion and the shock trails the planet. Observationally, we would detect a late egress, instead of an early 2 It is straightforward, however, to show that the kinetic term ingress. However, Figure 2 shows that for Pcrit,∗ < 1 d (WASP- and both thermal pressure terms are negligible relative to the mag- 12b), the relative azimuthal velocity ∆u is subsonic, therefore, no netic pressure terms. From the estimated planetary magnetic field 2 shock will be formed. (24 G), the magnetic pressure is pB ≃ Bp(rM ) /(4π) ≃ 8 × 4 Vidotto, Jardine & Helling

10 10 6 4x10 K Hydrostatic Corona Stellar wind 2x106 K 102 109 1x106 K

1 x (WASP-12b) * ) 1 1 0 -3 8 6 10 10 2x K 10

6 (km/s) r = 3.15 R

n (cm K u orb R (WASP-12b) * 0 7 10 10

4x1 0 6 = 3.15 R K orb R

6 -3 6 n = 1.5 x 10 cm -1 10 obs 10

1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5 4 Rorb (R*) Rorb (R*) Fig. 3.— Coronal density of an isothermal plasma for three dif- Fig. 4.— Coronal wind velocity of an isothermal wind for three ferent temperatures. diﬀerent temperatures. Filled circles represent the location of the sonic point (ur = cs) and dashed lines represent the regions of supersonic velocities, which, for T = 1 × 106 K, is outside the rewrite previous equation as range of orbital radii shown above.

Bc(Rorb) ≃ Bp(rM ), (10) simplest case of a thermally-driven wind which is not from which we can estimate an upper limit for the plan- magnetically channeled. The wind radial velocity ur is etary magnetic field intensity. derived from the integration of the differential equation For a stellar dipolar magnetic field, Bc(Rorb) = 3 ∂u ∂p GM∗ ∗ ∗ r B (R /Rorb) . Assuming that the planetary magnetic ρur = − − ρ . (12) 3 ∂r ∂r r2 field is also dipolar, then Bp(rM ) = Bp (Rp/rM ) . B∗ and Bp are the magnetic field intensities at the stellar and Figure 4 presents the wind velocity profile at different planetary surfaces, respectively. From Equation (10), we orbital radius up to a few stellar radii for three isothermal have wind temperatures. In the distance range Rorb < 4R∗, 3 3 the wind is still accelerating, and the radial velocities R∗/Rorb 1/3.15 B = B∗ = B∗ ≃ 2.4B∗, (11) are considerably smaller than the terminal velocities u∞ p −1 Rp/rM 1/4.2 achieved: u∞ ≃ 440, 670 and 1020 km s for T = 1, 2, and 4 × 106 K, respectively. The filled circles are the where we have used the observationally derived char- − sound speed of the wind: c ≃ 158 and 223 km s 1 for acteristics for the WASP-12 system: r = 4.2 R s M p T =2 and 4 × 106 K. Dashed lines represent the regions and R = 3.15 R∗. So far, tentative measure- orb of supersonic velocities. For the lowest-temperature case ments of the stellar magnetic field have not pro- (1 × 106 K), the wind becomes supersonic at a much vided significant detections (for theoretical predictions, larger radii, not shown in Figure 4. The vertical dotted see Christensen, Holzwarth, & Reiners 2009), but have line shown in Figure 4 represents the orbital radius of yielded an upper limit of 10 G in the longitudinal com- WASP-12b. ponent (Fossati et al. 2010b). Adopting this upper limit The density structure of the wind is derived from the as B∗, our model predicts a maximum planetary mag- conservation of mass, where nu r2 is a constant of the netic field of about 24 G. r steady-state wind. The dashed lines in Figure 3 present 4. STELLAR WIND the density profile for an isothermal wind, for three coronal temperatures. In contrast to the hydrostatic corona In this section, we investigate the scenario where the (solid lines), n does not depend on P∗. Except for the stellar magnetic field is not strong enough to confine lowest temperature adopted, where the density is well the coronal plasma, which expands in the form of stellar described by a hydrostatic density profile, the coronal wind. For a hot corona, the thermal pressure gradient base density n0 required for the formation of a shock is able to drive a wind (Parker 1958). We consider the in the wind case is larger than the one required by the −3 −2 2 hydrostatic corona case. Nevertheless, as in the static 10 dyn cm . The kinetic term ρc∆u /2 in Equation (9) is dependent on the period of the star, which for WASP-12 is unknown. scenario, a typical solar coronal density still allows for 2 −4 −2 For example, for P∗ = 26 d, ρc∆u /2 ≃ 4 × 10 dyn cm . the formation of a shock. 6 −4 −2 For T = 2 × 10 K, pc ≃ 4 × 10 dyn cm . To estimate In the stellar wind case, there are no azimuthal the thermal pressure of the planet, we use values derived from forces acting on the flow. Therefore, through conser- Murray-Clay et al. (2009) for planetary density (∼ 107 cm−3) and 3.5 −6 −2 vation of angular momentum of the particles leaving the temperature (10 K), which result in pp ≃ 4 × 10 dyn cm . 2 Because p is much larger than the aforementioned pressures, the star (2πR∗/P∗), the azimuthal velocity of the wind is B 2 magnetic terms in Equation (9) will dominate. uϕ,wind = 2πR∗/(P∗Rorb). This implies that ∆u for the Early UV Ingress in WASP-12b: Measuring Planetary Magnetic Fields 5

angle θ depends on the wind temperature for P∗ = 26 d. 90 While in the hydrostatic case, we found that an ahead- shock (θ = 0o) is always formed, the wind case requires 75 a very low temperature to form an ahead-shock. For higher wind temperatures, the shock normal gets closer to the line of centers between the planet and the star and o 60 Stellar Wind θ approaches 90 . ) o ( 45 5.

θ CONCLUSION Motivated by the recent observation on the light 30 curve asymmetry of the transit hot-Jupiter WASP-12b (Fossati et al. 2010a), we proposed a model where the interaction of the stellar plasma with the planet results in 15 the formation of a bow shock around the planet, which could explain the early transit ingress observed in the Hydrostatic corona near-UV. Although we know the orbital radius of the 0 planet, we do not know if at this radius the stellar mag- 1 1.5 2 2.5 3 3.5 4 T (MK) netic field is still capable of confining the hot gas of its corona, or if this gas is escaping in a wind. Therefore, Fig. 5.— Angle that the shock normal n makes to the relative we investigated the physical conditions of the external azimuthal velocity of the planet. ambient medium around the planet that could allow for wind case is the formation of such a shock. In either case, for plasma temperatures T . (4 − 5) × 1/2 2 6 GM∗ 2πR∗ 10 K we expect that a shock capable of compressing the ∆u = |uK − uϕ,wind| = − , (13) Rorb P∗Rorb plasma to the observed densities will form around the planet. In the case where the coronal plasma is confined, which is independent of the wind temperature or radial we showed that the geometry of this shock is independent velocity. For slow rotators, ∆u becomes almost indepen- of the plasma temperature. The shock forms ahead of the dent of P∗ and is given by ∆u ≈ uK. planet in its orbital path. In the unconfined (wind) case, 2 The condition for shock formation requires that (ur + the angle between the shock normal and the direction of 2 1/2 ∆u ) >cs, implying that planetary motion depends on the ratio of the radial wind speed and the azimuthal speed of the planet relative to 2 2πR∗/Rorb the stellar wind. Since the wind speed depends on the |P ∗| = , (14) crit, 2 2 2 1/2 temperature, the orientation of the shock is temperature- (u + ur − cs) K dependent. which, qualitatively, produces a similar result as the one If the planet’s orbit takes it through both regions of presented for the static case (Figure 2). This means confined plasma and also wind plasma, the orientation that the critical temperature derived in the previous and density of the shock may change, producing time- section (Equation (7)) is enhanced by a temperature dependent absorption and duration of transit. 2 5 −1 2 δT = urm/kB ≃ 8 × 10 (ur/(100 km s )) K, implying Given the large range of stellar rotation rates and coro- that for T . Tcrit + δT , a shock will be formed. nal/wind temperatures at which a shock is capable of In the wind case, Figure 4 shows that ur 6= 0 at the producing the densities estimated by LHV10, we con- orbital radius of WASP-12b, implying that the shock will clude that this is likely to be a common feature of tran- not form directly ahead of the planet, but at an interme- siting systems and may prove to be a useful tool in setting diate angle θ. Because θ depends on ur (Equation (1)), limits on planetary magnetic field strengths. For the case which depends on the temperature, the orientation of the of WASP-12b, we derived an upper limit for the planet’s shock is temperature-dependent. Figure 5 shows how the magnetic field of about 24 G.

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