Radio Emission from Red-Giant Hot Jupiters

0.89 MB·

by  Yuka Fujii

Draft version April 19, 2016
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
RADIO EMISSION FROM RED-GIANT HOT JUPITERS
Yuka Fujii1,2 David S. Spiegel3,4,5 Tony Mroczkowski6,7 Jason Nordhaus8,9
Neil T. Zimmerman10 Aaron R. Parsons11 Mehrdad Mirbabayi5 Nikku Madhusudhan12
6 1Earth-LifeScienceInstitute,TokyoInstituteofTechnology,Tokyo,152-8550,JAPAN
2NASAGoddardInstituteforSpaceStudies,NewYork,NY10025,USA
1
3Analytics&Algorithms,StitchFix,SanFrancisco,CA94103,USA
0 4Research&Development,SumLabs,NewYork,NY10001,USA
2 5AstrophysicsDepartment,InstituteforAdvancedStudy,Princeton,NJ08540,USA
6NationalResearchCouncilFellow
r 7NavalResearchLaboratory,4555OverlookAveSW,Washington,DC20375,USA
p
8DepartmentofScienceandMathematics,NationalTechnicalInstitutefortheDeaf,RochesterInstituteofTechnology,
A
Rochester,NY14623,USA
9CenterforComputationalRelativityandGravitation,RochesterInstituteofTechnology,Rochester,NY14623,USA
8 10SpaceTelescopeScienceInstitute,3700SanMartinDrive,Baltimore,MD21218,USA
1 11AstronomyDepartment,UniversityofCaliforniaBerkeley,Berkeley,CA,USAand
12AstronomyDepartment,UniversityofCambridge,UK
] Draft version April 19, 2016
P
E ABSTRACT
. Whenplanet-hostingstarsevolveoffthemainsequenceandgothroughthered-giantbranch,
h
the stars become orders of magnitudes more luminous and, at the same time, lose mass
p
at much higher rates than their main-sequence counterparts. Accordingly, if planetary
-
o companionsexistaroundthesestarsatorbitaldistancesofafewAU,theywillbeheatedup
r to the level of canonical hot Jupiters and also be subjected to a dense stellar wind. Given
t
s that magnetized planets interacting with stellar winds emit radio waves, such “Red-Giant
a Hot Jupiters” (RGHJs) may also be candidate radio emitters. We estimate the spectral
[ auroral radio intensity of RGHJs based on the empirical relation with the stellar wind as

well as a proposed scaling for planetary magnetic fields. RGHJs might be intrinsically
3
as bright as or brighter than canonical hot Jupiters and about 100 times brighter than
v
8 equivalentobjectsaroundmain-sequencestars. Weexaminethecapabilitiesoflow-frequency
2 radio observatories to detect this emission and find that the signal from an RGHJ may be
4 detectable at distances up to a few hundred parsecs with the Square Kilometer Array.
5
0 Keywords: planets and satellites: Jupiter — Sun: evolution — planetary systems — stars:
. evolution — stars: AGB and post-AGB — radio continuum: planetary systems
1
0
6 1. INTRODUCTION bation on light from the star, low-frequency plane-
1 taryradioemissionmightbeanarenawhereplanets
:
v Planets with strong magnetic fields may gener- are vastly brighter than their stars. This paper is
i ate radio and/or X-ray emission when interacting an exploration of the surprisingly diverse range of
X
with energetic charged particles. It is well known physicalprocessesthatleadtothisemissionandthe
r that Jupiter emits radio waves from its auroral re- prospects for detecting radio emission from planets
a
gionduetothecyclotron-maserinstability(e.g.Wu aroundgiantstarswithcurrentandnear-futurelow-
&Lee1979;Zarka1998;Treumann2006). Exoplan- frequency radio observatories.
ets could also generate radio emission through sim-
Observationsofradioemissionfromsolarsystem
ilar mechanisms, depending on their intrinsic mag-
planets imply an empirical relation: the auroral ra-
netic fields and the properties of surrounding plas-
diopowerisproportionaltotheinputsolarwinden-
mas, e.g. stellar wind particles and particles from
ergygoingintoeachplanetarymagnetosphere. This
Io-likemoons. Thisprocesscouldprovideanavenue
is commonly referred to as the “radiometric Bode’s
todiscoverplanetsthatareotherwiseextremelydif-
law” (Desch & Kaiser 1984; Zarka et al. 2001). Ex-
ficult to find — those orbiting highly evolved stars.
trapolating this scaling to exoplanets, radio emis-
Unlike traditional methods of discovering exoplan-
sionfromexoplanetarysystemshasbeenexamined.
ets, in which the planetary signal is a tiny pertur-
Farrell et al. (1999), Zarka et al. (2001), and Lazio
et al. (2004) estimated that a few of the known ex-
[email protected]
2
oplanetary systems may have radio flux of ∼1 mJy examined radio emission from known substellar-
level, due to the small orbital distance of the plan- mass companions around cool evolved stars. They
ets. Stevens (2005) gave improved estimates of the found that the low ionization fraction of the stellar
stellar mass-loss rate based on X-ray flux and re- windofevolvedstarssuppressestheirradioemission
evaluatedtheradiofluxofknownexoplanetarysys- and leads to weak radio emission. In addition, they
tems. Grießmeier et al. (2005) took account of the consideredascenarioinwhichpost-bowshockheat-
high stellar activity in the early stage of the plan- ing could produce ionized hydrogen atoms. This
etary system and proposed that the young system indicates a configuration of radio-wave generation
wouldbeagoodcandidateinwhichtosearchforra- very different from that of the solar system, and
dio emission. Grießmeier et al. (2007a,b) discussed the lower-limit estimate, at least, is well below the
the effects of the detailed properties of stellar wind detection limit.
in the proximity of the stars. They considered not
In this paper, we consider a further plausible
only the kinetic energy of the stellar wind but also
scenario, where the accretion of the massive stel-
themagneticenergyofthestellarwindandcoronal
lar wind onto the planet would emit UV and X-ray
massejection. Notethatinthesepapers,thescaling
photonsthationizethestellarwindinthevicinityof
relationsfortheplanetarymagneticfielddifferfrom
theplanet. Theionizedstellarwindparticleswould
paper to paper. In Reiners & Christensen (2010),
then interact with the planetary magnetic field in
the authors adopted a new scaling relation of plan-
thesame way asthesolar systemplanets do. Thus,
etary magnetic fields based on Christensen et al.
by extrapolating the “radiometric Bode’s law,” we
(2009). Jardine & Collier Cameron (2008) modeled
provide more optimistic estimates compared to the
the reconnection between the magnetic field of a
previous result. In addition, we introduce two ma-
close-in planet and that of the host star to obtain
joradvancesbeyondtheworkofIgnaceetal.(2010)
estimates that are not based on the “radiometric
that are related to the observability of radio emis-
Bode’s law.” Although observational searches for
sion from RGHJs. First, we estimate the plasma
theseradiosignaturesareunderway,nocleardetec-
frequency cut-off of the stellar wind, which turns
tion has been claimed (Bastian et al. 2000; George
out to be one of the major obstacles to detecting
&Stevens2007;Smithetal.2009;Stroeetal.2012;
radio emission from RGHJs. Second, we discuss
Hallinan et al. 2013; Murphy et al. 2015), while
the planetary parameter space to search for radio
there are some promising initial results (Lecavelier
emission, employing scaling laws for the planetary
des Etangs et al. 2013; Sirothia et al. 2014).
magneticfield, notingthatthesurveyofexoplanets
Whenstarslessthan∼8M evolveoffthemain around highly evolved stars is not complete.
(cid:12)
sequence, they evolve through the red-giant branch
In Section 2, we present the framework to ob-
(RGB) and the asymptotic-giant branch (AGB)
tainthefrequencyandthefluxofplanetaryauroral
phaseswheretheirradiiandluminositiesincreaseby
radio emission, and we describe our models for the
ordersofmagnitude. Jovianplanetsinorbitaround
stellar wind and planetary magnetosphere. Section
such stars can migrate inward or outward due to
3 presents the estimates of the spectral radio flux
the interplay between tidal torques and the mass-
of RGHJs and compares the predictions with what
loss process on the post-main sequence (Nordhaus
might be expected from canonical hot Jupiters as
etal.2010;Kunitomoetal.2011;Mustill&Villaver
well as those from Jupiter-twins. Section 4 gives
2012;Spiegel2012;Nordhaus&Spiegel2013). Dur-
the prospects for the signal detection with the cur-
ingthistime,suchplanetscanbetransientlyheated
tohot-Jupitertemperatures((cid:38)1000K)atdistances rent/future instruments. Estimates for the known
late-type (M-type) evolved stars are also included.
out to tens of AU, depending on the star’s mass;
Finally, Section 5 concludes the paper with a brief
such planets are termed “Red-Giant Hot Jupiters”
summary.
(RGHJs), though the term can refer to planets or-
biting either RGB and AGB stars (Spiegel & Mad-
husudhan 2012). Such planets are also subject to
2. MODEL
interactions with a massive (but slow) stellar wind,
as the mass-loss rate of evolved stars is significant,
In this section, we describe our framework to
ranging from ∼10−8 M yr−1 to ∼10−5 M yr−1
(cid:12) (cid:12) estimate radio emission from RGHJs. First, we in-
withthehighestvaluesforAGBstars(e.g.,Reimers
troduce our scheme to compute the frequency and
1975;Schild1989;Vassiliadis&Wood1993;Sch¨oier
intensityofplanetaryradioemissioninSections2.1
& Olofsson 2001; van Loon et al. 2005). On the
and 2.2, respectively. Then, two ingredients for the
assumption that the radio emission is correlated
framework — the strength of the planetary mag-
withthestellarwind,planetarycompanionsaround
netic field and the properties of the stellar wind —
evolved stars could also generate bright radio emis-
will be presented in Sections 2.3 and 2.4, respec-
sion.
tively. In Section 2.5, we consider the ionization
Based on this speculation, Ignace et al. (2010) around the planets, which is a crucial factor to de-
terminetheefficiencyoftheinteractionbetweenthe
3
planetary magnetosphere and the stellar wind. Ωisthesolidangleoftheemission,l isthedistance
between the target and the Earth, and ∆ν is the
frequency bandwidth.
2.1. Frequency of Radio Emission
We estimate the radio emission of exoplanets,
As ionized electrons flow along planetary mag-
P , simply by scaling the Jovian auroral radio
radio
netic field lines, auroral radio waves are emitted at
emission, P , with the input energy from stel-
radio,J
the local cyclotron frequency. The upper limit is
lar wind, in the same manner as Grießmeier et al.
aroundthecyclotronfrequencyoftheplanetarysur-
(2005, 2007a,b). The scaling is based on the empir-
face magnetic field, ν :
cyc,max ical/apparently good correlation between the radio
(cid:18) (cid:19) emission intensity of solar system planets and the
eB B
ν = ≈28 MHz (1) inputkineticenergy, P , orthemagneticenergy
cyc,max 2πm c 10 G inp,k
e of the solar wind, P , (the “radiometric Bode’s
inp,m
where B is the strength of the magnetic field at the law”; Desch & Kaiser 1984; Zarka et al. 2001), i.e.,
planetarysurface,eandm arethechargeandmass
e
ofelectrons,respectively,andcisthespeedoflight. Pradio∝Pinp (6)
P =m nv3·πr2 , (7)
Radio emission from an exoplanet is observable inp,k p mag
fromtheground(onEarth)onlywhenitsfrequency P =(B2 /8π)v·πr2 , (8)
inp,m (cid:63)⊥ mag
isgreaterthanboththeplasmafrequencyofEarth’s
ionosphere ν⊕ and the maximum plasma fre- where m is the proton mass, n is the number den-
plasma p
quency along the line of sight νlos : sity of the stellar wind, v is the relative velocity
plasma
of the stellar wind particles to the planet, B is
(cid:63)⊥
ν >ν⊕ and ν >νlos (2) the interstellar magnetic field perpendicular to the
cyc,max plasma cyc,max plasma
stellar wind flow, and r is the distance from the
mag
The plasma frequency may be expressed as
centeroftheplanettothemagneticstand-offpoint,
(cid:115) describedbelow. Itisnotclearfromobservationsof
n e2
ν = e (3) the solar system planets which of the above two re-
plasma πm lations (Eqs. 7 and 8) most accurately captures the
e
“true” relationship between input ingredients and
(cid:16) n (cid:17)1/2
=8979 Hz× e . (4) output radio power (Zarka et al. 2001). In this pa-
cm−3 per, we assume that the radio emission scales with
In the Earth’s ionosphere, the electron number the input kinetic energy (i.e., that P ∝ P ,
radio k,inp
density is less than 106 cm−3, which implies that as per Eq. 7) and consider the possible effects of a
νp⊕lasma ∼<10MHz. Alongthelineofsight,themaxi- dense stellar wind.
mumplasmafrequencyνlos istypicallygoverned
plasma NotethatifEq.8isactuallythebetterpredictor
by the density of stellar wind particles around the ofradiopower,itisdifficulttoestimatetheemission
planet, which will be specified in Section 2.4 below. from RGHJs at this point, because magnetic fields
of evolved stars are not well constrained for highly
Another factor that affects the radio emission
evolvedstars;mostobservationssetonlyupperlim-
is the plasma frequency at the site of radio-wave
its (e.g., Konstantinova-Antova et al. 2010, 2013;
generation. This is because the cyclotron-maser in-
Petitetal.2013;Tsvetkovaetal.2013;Auri`ereetal.
stability as a mechanism to generate intense radio
2015). In the case of the M-type giant EK boo,
emission is efficient when the in situ plasma fre-
a surface magnetic field ∼0.1-10 G has been mea-
quencyissmallincomparisontothelocalcyclotron
sured. Inthisparticularcase,giventhelargestellar
frequency (Treumann 2006). Future work will be
radius(R ∼210R ),themagneticmomentmaybe
required to estimate the local plasma density and (cid:63) (cid:12)
about 103× larger than that of the Sun and there-
examine this condition. In this paper, we assume
foremayalsoincreasetheplanetaryradioemission.
that this condition is satisfied at least at some re-
However, due to our present ignorance about the
gion along the poloidal magnetic field lines.
strengthofmagneticfieldsofevolvedstars,weleave
themagneticmodelofradioemissionforRGHJsfor
2.2. Flux of Radio Emission
future work.
Theauroralradiospectralfluxofexoplanetsob- Inreality,thetotalpowerofJovianauroralemis-
served at the Earth, F , can be expressed by: sion varies greatly over time. The average is of the
ν
order of 1.3×1010 W. During highly active peri-
P
F = radio (5) ods, it averages more like 8.2 × 1010 W, and the
ν Ωl2∆ν emission can reach powers as high as 4.5×1011 W
whereP istheenergythatisdepositedasauro- during peak activity (Zarka et al. 2004). Here, we
radio
ralradioemissionoftheconsideredfrequencyrange, employP =2.1×1011 Wasacanonicalvalue,
radio,J
4
following Grießmeier et al. (2005, 2007b). Christensenetal.2009foracomprehensivedescrip-
tion). This scaling law is based on the assumption
The magnetic stand-off point is where stellar
that the ohmic dissipation energy is a fraction of
windrampressureandplanetarymagneticpressure
the available convected energy and was found to
are in approximate balance:
be in good agreement with both the numerical ex-
B2 (cid:18)r (cid:19)−6 periments (over a wide parameter space) and with
mpnv2 ∼ 8π Rmapg . (9) kanndowFn oabrjeecatsssufrmomedthtoe Ebaertchontostastnatrsf.orHethree, fboohdm-
ies considered in this paper. Dipole magnetic field
The outflow ram pressures from heated planets are
strength at the planetary surface, denoted by B, is
negligiblecomparedtothesepressures,asdiscussed
then scaled by
in Appendix A. Therefore,
rmag ∼Rp(8πmpn)−1/6v−1/3B1/3 (10) B ∝Bdynamo(cid:18)rdyRnamo(cid:19)3 . (12)
The radius obtained for Jupiter from this equal- p
ity is about half of the actual magnetospheric ra- where r is the radius of the outer boundary
dynamo
dius (Grießmeier et al. 2005). To estimate r for of the dynamo region.
mag
RGHJs, we scale this radius according to the pa-
The scaling law for B , Equation (11), is
rameter dependence of equation (10). dynamo
reasonable only for rapidly rotating objects. Un-
For exoplanets, we assume that the solid angle like canonical hot Jupiters, RGHJs are not tidally
of the emission (Ω) is the same as that of Jupiter. locked to their host stars, so they probably have
In reality, the solid angles of auroral radio emission the rapid rotation needed for Eq. 11 to be a use-
from Jupiter, Saturn, and Earth are ∼1.6, ∼6.3, ful ansatz (Spiegel & Madhusudhan 2012). In this
and ∼3.5 steradians, respectively (Desch & Kaiser paper, we assume that RGHJs indeed are rapidly
1984), which are on the same order and will not rotating so that they generate planetary magnetic
significantly affect our order-of-magnitude estimate fields through the same mechanism as Jupiter.
of radio emission. The bandwidth, ∆ν, is assumed
In order to evaluate ρ and q , we need
to be proportional to the representative frequency dynamo o
a model of the internal planetary structure. We
of the emission, which is the cyclotron frequency,
consider Jupiter-like gaseous planets and assume
following Grießmeier et al. (2007b).
that the planetary radius is constant at R =R ,
p p,J
as numerical calculations show that the radii of
2.3. Assumptions for Planetary Magnetic Field gaseous planets over the range of 0.1M < M <
p,J p
10M (with core mass less than 10%) converge to
p,J
In order to obtain the frequency and the inten- 0.8R <R <1.2R within1Gyr(Fortneyetal.
p,J p p,J
sity of the radio emission, we need to compute the 2007; Spiegel & Burrows 2012, 2013). For the den-
strength of the magnetic field at the planetary sur- sity profile, we follow Grießmeier et al. (2007b) and
face, B. We do so simply by scaling the Jovian assume a polytropic gas sphere with index n = 1,
magnetosphere at the surface BJ ∼10 G according which results in:
totheplanetarymassandage,basedonthescaling (cid:104) (cid:105)
relation described below. ρ[r]=(cid:18)πMp(cid:19)sin πRrp . (13)
(cid:16) (cid:17)
Several scaling relations for planetary magnetic 4Rp3 π r
field strength have been proposed (e.g. Busse 1976; Rp
Russell 1978; Stevenson 1979; Mizutani et al. 1992; We determine the radius of the outer boundary
Sano 1993; Starchenko & Jones 2002; Christensen of the dynamo region, r , by assuming that
dynamo
& Aubert 2006; Christensen et al. 2009), which are the hydrogen becomes metallic when ρ(r) exceeds
summarized and compared with numerical simula- the critical density ρ = 0.7g/cm3 (Guillot 2006;
crit
tions in Christensen (2010). We employ the scaling
Grießmeieretal.2007b). Thedensityofthemetallic
law proposed in Christensen et al. (2009) and used
core,ρ isobtainedbyaveragingthedensityin
dynamo
inReiners&Christensen(2010)toexploretheevo-
the core. In the case of Jupiter, r =0.85R
dynamo,J J
lution of planetary magnetic fields:
and ρ =1.899 g/cm3.
dynamo,J
B2 ∝f ρ1/3 (Fq )2/3, (11)
dynamo ohm dynamo o The scaling of convected heat flux at the outer
boundary, q , is obtained by dividing the age-
where B is the mean magnetic field in the o
dynamo
dependentnetplanetaryluminosity,L ,bythesur-
dynamo region, f is the ratio of ohmic dissipa- p
ohm
tion to the total dissipation, ρ is the mean face area of the core region, i.e., 4πr2 . The
dynamo dynamo
density in the dynamo region, F is an efficiency time-dependent luminosity is taken from equation
factor of order unity, and q is the convected flux (1) of Burrows et al. (2001) (see also Marley et al.
o
at the outer boundary of the dynamo region (see 2007). Ignoring the relatively weak dependence on
5
average atmospheric Rosseland mean opacity leads lar wind particles falling onto the planet, depends
to: on the stellar wind velocity, the infall velocity (i.e.,
(cid:18) t (cid:19)−1.3(cid:18)M (cid:19)2.64 the acceleration due to planet’s gravitational field),
L ∼6.3×1023 erg p . and the planetary orbital velocity. All three of
p 4.5 Gyr MJ these terms might be of the same order of mag-
(14) nitude: as described above, v is ∼30 km/s; the
sw
Therefore, we have infall velocity from the planetary gravity is ∼10-
25 km/s depending on the planetary mass (rang-
(cid:18) t (cid:19)−1.3(cid:18)M (cid:19)2.64(cid:18) r (cid:19)−2
qo ∼qo,J 4.5 Gyr MpJ rddyynnaammoo,J . i1n0g−fr3o0mk1mMs−Jt1o1d0eMpeJn)d;ianngdonthtehoerboritbailtavledloicstitayncies
(15) (ranging from 1−5 AU). In the next section, we
simply consider
2.4. Assumptions for Stellar Wind v
∼10−1 (19)
v
(cid:12)
The other key ingredient for radio emission is
thestellarwind. Thenumberdensityofparticlesin a fiducial value for the normalization.
the stellar wind, n, can be expressed as
M˙ 2.5. Ionization of Stellar Wind Particles Around
n= (cid:63) , (16)
the Planet
4πa2m v
p sw
where M˙ is the stellar mass-loss rate, a is the As discussed in Ignace et al. (2010), as stars
(cid:63)
orbital distance from the star, m is the proton evolve,theionizationfractionofthestellarwinddi-
p
mass, and v is the velocity of the stellar wind. minishes to the order of ∼10−3 (Drake et al. 1987).
sw
For the solar wind, M˙ ∼ 2 × 10−14M /yr and Since only charged particles interact with a plane-
(cid:12) (cid:12)
v ∼400 km/s (e.g., Hundhausen 1997). tarymagneticfield, thissuggestsinefficientinterac-
sw
tion with the planetary magnetosphere and hence,
The mass-loss rate of red giants is typically low input energy for radio emission. However, for a
M˙ ∼ 10−8 − 10−7M /yr (Reimers 1975), and highly evolved star, the velocity of the stellar wind
(cid:63) (cid:12)
the rate can be as high as 10−5M /yr during the eventually becomes slower than the escape veloc-
(cid:12)
AGB phase (Schild 1989; Vassiliadis & Wood 1993; ity of the planetary companion and hence, stellar
Sch¨oier & Olofsson 2001; van Loon et al. 2005). windparticleswillaccreteontotheplanets. Spiegel
Therefore, we have &Madhusudhan(2012)consideredBondi–Hoyleac-
cretion where the accretion radius is
M˙
M˙(cid:12)(cid:63) ∼106−109. (17) Racc=2Gv2Mp (20)
rel
(cid:18)M (cid:19)(cid:16) v (cid:17)−2
as tThheestsaterlleavrolwviensd. vTehloecwityin,dvsvwe,lobceictyomisestyspmicaalllelyr ∼4 RJ Mp 30 kmrels−1 . (21)
J
of the order of the escape velocity at a distance of
TheaccretionluminosityL andtemperatureT
several times the stellar radius (Suzuki 2007), i.e., acc acc
(cid:112) are
∼ 2GM /R(cid:48), where R(cid:48) is several times R . For
(cid:63) (cid:63) (cid:63) (cid:63) (cid:32) (cid:33)
vaswsta∼r 3w0itkhmr/asd.iuTshRer(cid:63)efo=re,10a0soRl(cid:12)ar,-mthaisssrreesdulgtsianint Lacc ∼1025 ergs−1 10−8MM˙(cid:63) /yr
with radius R = 100R produces a stellar wind (cid:12)
(cid:63) (cid:12)
that is slower by an order of magnitude than that (cid:18)M (cid:19)3(cid:18)M (cid:19)−2
of the Sun, v . × p (cid:63) . (22)
(cid:12) M M
J (cid:12)
Based on equation (17), the number density of (cid:18)M (cid:19)
the stellar wind (equation 16) is normalized as fol- Tacc ∼2×105 K Mp . (23)
lows: J
(cid:16) a (cid:17)−2 Theaccretionontoplanets,therefore,leadstoemis-
n=1.8×106 cm−3× sion of UV/X-ray photons whose characteristic en-
5AU
×(cid:32) M˙(cid:63) (cid:33)(cid:18) vsw (cid:19)−1 . (18) edrrgoyge∼n,kBETRaycdcbeerxgc=eed13s.6theeVio(nλiz=ati9o1n.2ennmer)g.yTohfehrye--
10−8M /yr 30 km/s fore, UV/X-ray radiation from accretion will create
(cid:12)
a local ionized region around the planet.
The velocity term in equation (7), which is the Letusconsidertheionizationprofilearoundthe
relative velocity between the planet and the stel- planet. Wesupposeastatewheretheionizationand
6
recombination rates are in equilibrium. Denoting or it can (ii) go entirely into the kinetic energy of
the ionization fraction by x, the equilibrium state oneelectronandnotatallintothatoftheother(of
at a distance r from the planet may be represented course, any split in between these extremes is pos-
by sible, too). Note that conservation of momentum
andenergyimplythattheprotonwillnotacquirea
N˙X e−τn(1−x)σ [E ]=(nx)2β[T ] (24) significant fraction of the energy of the collision.2
4πr2 H photon e
Scenario (i) implies a cascade where an elec-
(cid:90) r tron with energy E ionizes an atom, producing
τ = n(1−x)σ [E ]dr, (25) in
H photon twoelectrons(anionizingelectronplusthereleased
RJ electron) with energy (E −E )/2 for each.
in Rydberg
whereN˙X isthesourcerateofthephotonsthatcan For example, in an idealized case with Mp =10MJ,
ionize hydrogen, Ephoton is the energy per photon, kBTacc ∼ 172eV ∼ 12ERydberg, a photoionization
and σ is the cross-section of H atoms for X-ray couldproduceanelectronwithenergyof(12−1)=
H
photons. Per Verner et al. (1996), σH scales as 11ERydberg, then a second ionization by that elec-
tron would produce two photons with energy of
(cid:18) E (cid:19)−3 (11−1)/2=5E , and the third ionization by
σ [E ]∼6.3×10−18 cm2· photon , Rydberg
H photon E thesetwophotonswouldproducefourphotonswith
Rydberg
energy of (5−1)/2 = 2E , etc. The cascade
(26) Rydberg
can proceed to the fourth order at the maximum.
where β[T ] is the “class B” recombination coef-
e
ficient as a function of the electron temperature, Under scenario (ii), the cascade proceeds with
2T.e6. ×W10e−1a3docpmt3t/hseec,vainluethaetfoTlelow∼ing10(4PeKq,uiβgno∼t tkhineeitniciteianlerEgiyn Eelec−troEn leadingantodaannoetlheecrtreolnecwtriotnh
in Rydberg
et al. 1991); when the electron temperature is var- with zero kinetic energy. Clearly, this cascade can
ied from 103 to 105 K, β varies 1.5×10−12−3.2× produceamaximumtotalofE /E freeelec-
in Rydberg
10−14 cm3sec−1. trons.
The source rate is obtained by counting the Inreality,notallreleasedelectronsresultinfur-
numberofphotonswithenergyexceedingERydberg, therionizationinteractions. Ifξ representsthefrac-
which is approximately given by dividing the X-ray tion of released electrons that proceed to the next
accretion luminosity by the characteristic photon ionization, thenthenumberofionizedatomsN re-
i
energy produced: leased through this cascade (i) is
L
N˙X ∼ k Tacc , (27) Ni=(1−ξ)+2ξ(1−ξ)+4ξ2(1−ξ)+8ξ3
B acc =1+ξ+2ξ2+4ξ3. (29)
where k is Boltzmann’s constant. As a result,
B
Alternatively, cascade (ii) leads to
(cid:26)
4×1035 sec−1 (M =M )
N˙ ∼ p J . (28)
X 4×1037 sec−1 (M =10M ) 1−ξk
p J
N =
i 1−ξ
For accretion onto very massive planets, the =1+ξ+ξ2+···+ξk−1, (30)
characteristic photon energy is so high that the
released energetic electrons may also ionize other wherek ≡E /E isthemaximumnumberof
in Rydberg
atoms in the vicinity. The cross-section1 of hydro- ionizationsforthegiveninitialelectronenergy. This
gen for electrons is ∼4×10−17 cm2 (Fite & Brack- limitleadstoavalueforN thatisnotdramatically
i
mann1958),whichimpliesameanfreepathforion- different from that of limit (ii). In Appendix B,
ized electrons of ∼2R in the surrounding medium. we estimate the Møller scattering cross-section and
J
As a result, nearly 2/3 of released energetic elec- showthatcascade(ii)—unequalrecoilenergies—
tronswillionizeahydrogenatomwithin∼2R ,and is probably more realistic.
J
more than 99% will ionize a hydrogen atom within
∼10RJ. Ultimately, N˙X in equation (24) is replaced by
In principle, photons with energy Ephoton have N˙X →NiN˙X. (31)
the potential to ionize E /E hydrogen
photon Rydberg
atoms. Toaccountforthis,weconsidertwolimiting Fora10M planet,N isprobablyapproximatelyin
J i
possibilities: After an ionizing collision, the energy the range 5—10.
can (i) be split evenly between the two electrons,
2 Were it otherwise, we would have to take into account
1 This is close to the geometric cross-section of the Bohr whatfractionofarubberball’skineticenergyisimpartedto
radius. thekineticenergyoftheEarthwhenbouncingaball.
7
a=5AU, 10−8M /yr which gives
1.0 ⊙
rstromgrenrstromgren (cid:26)67 R (M =M )
r ∼ J p J (35)
0.8 stromgren 310 R (M =10M )
J p J
n
o
cti for a planet 5 AU from a red giant host. The
a 0.6
n fr Mp =MJ Sintdr¨oicmatgerdeninraFdiigiuforre11-.MJand10-MJplanetsarealso
atio 0.4 Mp =10MJ
z Note that this is a very interesting and different
oni regimeoftheStr¨omgrenspherefromthecommonly-
i 0.2 ag ag considered case (photoionization around O/B-type
m m
r r stars). Around RGHJs, the Str¨omgren sphere does
not delineate a sharp edge of ionization, because
0.0
100 101 102 103 of the smaller source rate and smaller photon-
hydrogen cross-section (in this case parameter “a”
distance from planet [R]
J in equation 13 of Str¨omgren 1939 is not small) on
Figure 1. Profile of ionization fraction measured from the account of X-ray photons interacting more weakly
surfaceoftheplanet,duetoUV/X-rayfromtheaccretionof with neutral hydrogen than UV photons near the
stellar wind onto the planet. Solid lines show the solutions
ionization limit. The Str¨omgren radii are indicated
withoutthecorrectionforthesecondaryionizationbyionized
electrons,andthedashedlineshowsthesolutionfora10MJ byverticalarrowsinFigure1. Crucially, X-rayand
planettakingthecorrectionintoaccountwithefficiencyfac- UV emission from accretion onto the planet should
tor Ni =6. The vertical arrows show the Str¨omgren radius ionize a significant fraction of the incoming stellar
estimated simply using equation (35). The dotted vertical
wind, thereby allowing radio waves to be generated
lines indicate the location of the magnetic stand-off radii,
rmag,obtainedusingequation(40)below. from this interaction.
Inreality,thetemperatureandluminositybased
ontheBondi–Hoyleaccretion(equations22and23)
describes the situation only approximately. Pre-
The ionization fraction x as a solution of equa- cisely, only the neutral portion of the stellar wind
tion (24) is shown in Figure 1. When τ ∼ nσHr canaccreteontotheplanetwithoutinteractingwith
is much smaller than unity and thus the e−τ term the planetary magnetic field, and the ionized por-
can be ignored (as is the case for Mp =10MJ), the tion would lose some energy at the bow shock be-
solution is simply fore it accretes. On the other hand, the ionized
(cid:112) plasma interacting with the magnetic field could
−1+ 1+4C[r]
haveacross-sectionlargerthantheBondi–Hoyleac-
x[r]= (32)
2C[r] cretion radius. The detailed electromagnetic struc-
4πnβ[T ]r2 turearoundRGHJsthereforerequireselaboratenu-
C[r]≡ e . (33) merical simulations that are beyond the scope of
N˙σH[Ephoton] thispaper. Inthefollowingsections,weaimtogive
order-of-magnitude estimates of radio emission, ob-
The solid lines show the ionization fraction corre-
servability, etc., which ought to be robust with re-
sponding to no additional ionization by electrons
specttouncertaintiesinthedetailsoftheionization
(i.e., ξ = 0, or N = 1) and the dashed line shows
i process.
the solution with N =6 for a 10M planet. In the
i J
figure,theverticallinesshowthemagneticstand-off
radius obtained using equation (40) below. While
the photon rate is larger for more massive planets, 3. ESTIMATES
the strong dependence of the cross-section on pho-
ton energy (equation 26) leads to a decrease in the 3.1. Planetary Magnetic Field and Frequency of
radius of the ionized region. Nevertheless, a sub- Radio Emission
stantialamountofionizedplasmaisexpectedaround
the magnetic stand-off radius, despite the initially Figure2showsthecomputedradius(rc)andav-
low ionized fraction of the stellar wind. erage density (ρc) of the dynamo region as a func-
tion of planetary mass, as well as the heat flux
The extent of the ionized region may also at the outer boundary of the core (q ), the esti-
o
be roughly estimated as the Str¨omgren radius matedstrengthoftheplanetarymagneticfield(B),
(Str¨omgren 1939): andthecorrespondingcyclotronfrequency(ν )as
cyc
functionsofplanetarymassandtheage. Substitut-
(cid:32) (cid:33)1/3
3 N˙ ingequation(15)intoequation(12),andgiventhat
rstromgren = 4πn2Xβ , (34) rdynamo does not change significantly, the magnetic
8
1.0 future radio observatories including the Giant Me-
trewave Radio Telescope (GMRT), Low-Frequency
]
RJ Array (LOFAR), Hydrogen Epoch of Reionization
[ 0.8 Array (HERA), Square Kilometer Array (SKA),

o
m and potential upgrades to the Very Large Array
a
yn (VLA) (see Section 4.3).
rd 0.6
Since ν >ν⊕ , the radio emission will
20 cyc,max plasma
] not be hindered by Earth’s ionosphere cut-off. On
3m 15 the other hand, it may experience opacity due to
/c the plasma of the stellar wind particles around the
g
[ 10 planet. The maximum plasma frequency along the

mo line of sight, νlos , corresponds to that in the
yna 5 vicinity of the ppllaasnmeat, if the planet is on the near
d
ρ side of its star to the Earth. Therefore, substitut-
0
0 2 4 6 8 10 12 ing equation (18) to ne in equation (4),
M [M ]
p J (cid:32) (cid:33)1/2
] 106 νlos ∼8979 Hz× M˙(cid:63) ×1 cm3 (37)
2m plasma 4πa2mpvsw
/c 104
ec (cid:16) a (cid:17)−1(cid:18) v (cid:19)−1/2
rg/s 102 =12 MHz× 5 AU 30 ksmw/s
e
[
qo100 (cid:32) M˙ (cid:33)1/2
103 × 10−8M(cid:63) /yr . (38)
(cid:12)
]
[BG102 We can see emission only from where νcyc,max >
νlos . Thedetectableparameterspacewillbepre-
101 plasma
sented in more detail in the next section.
] 103
z
H
[νMcyc,max 110012 15MM0JMJJ The3m.2.agFnCeluotximcopsaftarRinsGdo-HnoffJwriRtahaddiCuioasn(EeomqnuiicsaastiliooHnnJ1isn0)may
101008 109 1010 be written as follows using the stellar mass-loss
rate:
age [yr]
Figure 2. Upperpanels: radiusandaveragedensityofthe (cid:18) B (cid:19)1/3(cid:16) a (cid:17)1/3
r =r
dynamoregionasafunctionofplanetarymass. Lowerpan- mag mag,J B 5.2AU
els: evolutionsofheatfluxattheouterboundary,planetary J
meretasdg)w.nietthicvfiaeryldin,ganmdamssaesxi(mMuJm: cbylucelo,t5roMnJf:regqrueeenn,cya,nfdor10pMlanJ-: ×(cid:32)MM˙˙(cid:12)(cid:63)(cid:33)−1/6(cid:18)vv(cid:12)(cid:19)−1/6 (39)
field is approximately:
The typical value for RGHJs is found by substi-
(cid:18)M (cid:19)1.04(cid:18) t (cid:19)−0.43 tuting relevant values for stellar wind parameters
B ∼B p (36) described in Section 2.4:
J M 4.5 Gyr
J
(cid:18) B (cid:19)1/3(cid:16) a (cid:17)1/3
ustnrdenergtthhiosfmJuopdietle,r;wihnertehisBpJaipsert,hewemraogungehtilcy cfioenld- rmag∼14RJ BJ 5AU
suiedseragBreJe∼wit1h0RGe.ineRresa&sonCahbrliys,tetnhseenre(s2u0lt1a0n),twvahlo- ×(cid:32) M˙(cid:63) (cid:33)−1/6(cid:18) v (cid:19)−1/(640)
adoptedthesamescalinglawfortheplanetarymag- 10−8M /yr 10−1v
(cid:12) (cid:12)
neticfield;weshowthisfigurejustforcompleteness.
NotethatthecyclotronfrequencyofJovianplanets
where we employ r =84 R (Joy et al. 2002).
mag,J J
typically falls between 10 MHz and 1 GHz. In this
regime, there are a number of current and near- Substituting equation (39) to equation (7), the
9
radio flux in Jy
from a target at 100 pc, age 4.5 Gyr
2
10
RG (M˙ =10 8M /yr)
−
U] ⊙ 1e-07
A
[
a

e
c
n
1
a 10
st 1e-06
i
d

l
a
t
i
b
r
o
1e-05
0
10
2
10
AGB (M˙ =10 5M /yr)
−
U] ⊙ 1e-05
A
[
a

e
c
n
1
a 10
st 1e-04
i
d

l
a
t
i
b
r
o
1e-03
0
10
4
10
]
z
H
3
M 10
[
x
ma 102
c,
y
c
ν 1
10
0 1 2
10 10 10
planetary mass M [M ]
p J
Figure 3. Radio flux in unit of Jy from a planetary companion to a red giant with mass loss rate 10−8M(cid:12)/yr (top) and
thattoanAGBstarwithmasslossrate10−5M(cid:12)/yr(bottom). Thesystemsarelocatedat100pcaway. Thedoubly-hatched
regions show the parameter spaces where the planetary radio emission would not be observable at all because the maximum
frequency of the emission (cyclotron frequency at the planetary surface, νcyc) is below the plasma frequency cut-off, νplolassma.
Theregionshatchedwithverticallinesshowtheparameterspaceswherethefrequenciesofbulkradioemissionisbelowνlos ,
plasma
i.e.,νplolassma>0.1νcyc.
10
scalingoftheradioemissionisexpandedasfollows: (cid:18) B (cid:19)−1/3(cid:16) a (cid:17)−4/3
×
B 5 AU
P =m nv3·πr2 (41) J
k,inp p (cid:18) Bma(cid:19)g 2/3(cid:16) a (cid:17)−4/3 ×(cid:32)M˙(cid:63)(cid:33)2/3(cid:18) v (cid:19)5/3 (47)
=Pk,inp,J B 5.2 AU M˙(cid:12) v(cid:12)
J
(cid:32)M˙ (cid:33)2/3(cid:18) v (cid:19)5/3 (for a Jupiter-twin)
× (cid:63) (42) (cid:18) d (cid:19)−2
M˙(cid:12) v(cid:12) ≈0.70×10−5Jy
100 pc
(cid:18) B (cid:19)−1/3(cid:16) a (cid:17)−4/3
We may compare the radio emission power of ×
RGHJs at 5 AU with that of canonical hot Jupiters BJ 5 AU
seeqtuaatti0o.n05(4A2U).caInnobrederre-tnooprmerafolirzmedthasisfcoollmowpsa:rison, ×(cid:32) M˙(cid:63) (cid:33)2/3(cid:18) v (cid:19)5/3 (48)
10−8M /yr 10−1v
(cid:12) (cid:12)
(cid:18) B (cid:19)2/3(cid:16) a (cid:17)−4/3
P ≈140 P (for RGB stars’ companions)
k,inp k,inp,J B 5 AU
J (cid:18) d (cid:19)−2
×(cid:32) M˙(cid:63) (cid:33)2/3(cid:18) v (cid:19)5/3 (43) ≈0.70×10−3Jy 100 pc
10−8M(cid:12)/yr 10−1v(cid:12) (cid:18) B (cid:19)−1/3(cid:16) a (cid:17)−4/3
×
(for RGB stars’ companions) B 5 AU
J
≈14000 Pk,inp,J(cid:18)BB (cid:19)2/3(cid:16)5 AaU(cid:17)−4/3 ×(cid:32) M˙(cid:63) (cid:33)2/3(cid:18) v (cid:19)5/3 (49)
J 10−5M /yr 10−1v
(cid:12) (cid:12)
×(cid:32) M˙(cid:63) (cid:33)2/3(cid:18) v (cid:19)5/3 (44) (for AGB stars’ companions)
10−5M(cid:12)/yr 10−1v(cid:12) (cid:18) d (cid:19)−2
≈2.4×10−5Jy
(for AGB stars’ companions) 100 pc
(cid:18) B (cid:19)2/3(cid:16) a (cid:17)−4/3 (cid:18) B (cid:19)−1/3(cid:16) a (cid:17)−4/3
≈490 Pk,inp,J BJ 0.05 AU × BJ 0.05 AU
×(cid:32)M˙(cid:63)(cid:33)2/3(cid:18) v (cid:19)5/3 (45) ×(cid:32)M˙(cid:63)(cid:33)2/3(cid:18) v (cid:19)5/3 (50)
M˙(cid:12) v(cid:12) M˙(cid:12) v(cid:12)
(for canonical hot Jupiters) (for canonical hot Jupiters)
Here, we have normalized the magnetic field
strength of canonical hot Jupiters with B , consid- Thus, RGHJs are expected to be intrinsically as
J
eringtheuncertaintyofthemagneticfieldsoftidally bright as the closest hot Jupiters. Compared with
locked planets. Note that some models of plan- the equivalent Jupiter-like planets around main se-
etary magnetic field strength predict that tidally quencestars,themassivestellarwindoflateredgi-
locked planets have weaker magnetic fields due to ants can increase the radio emission from planetary
theirslowrotation(e.g.Grießmeieretal.2004). Al- companionsbymorethantwoordersofmagnitude,
thoughtheorbitalvelocityofcanonicalhotJupiters which allows us to explore 10 times more distant
has been ignored in equation (45), the Keplerian systems,i.e.,1000timesmorevolume. Thisatleast
velocity at 0.05 AU around a solar-mass star is partially compensates for the small population of
∼130kms−1,whichresultsinonlya<10%increase evolved stars.
of the relative velocity.
Equation (48) gives a spectral flux density one
Using the expressions above, we find that the to two orders of magnitude smaller than the pre-
radio spectral flux density observed at the Earth diction of equation (5) in Ignace et al. (2010) if we
is: assume the same set of fiducial values and full ion-
ization. This is primarily because their formula-
P
F = radio (46) tion did not incorporate the effect of a compressed
ν Ωd2νcyc planetary magnetosphere due to a massive stellar
(cid:18) d (cid:19)−2 wind, while the scaling law for the planetary mag-
≈5.2×10−8Jy netic field strength is also different.
100 pc