A&A 475, 359-368 (2007)
J.-M. Grießmeier1 - P. Zarka1 - H. Spreeuw2
1 - LESIA, Observatoire de Paris, CNRS, UPMC, Université Paris Diderot; 5 Place Jules Janssen, 92190 Meudon, France
2 - Astronomical Institute "Anton Pannekoek'', Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
Received 3 March 2007 / Accepted 18 August 2007
Context. Close-in giant extrasolar planets ("Hot Jupiters'') are believed to be strong emitters in the decametric radio range.
Aims. We present the expected characteristics of the low-frequency magnetospheric radio emission of all currently known extrasolar planets, including the maximum emission frequency and the expected radio flux. We also discuss the escape of exoplanetary radio emission from the vicinity of its source, which imposes additional constraints on detectability.
Methods. We compare the different predictions obtained with all four existing analytical models for all currently known exoplanets. We also take care to use realistic values for all input parameters.
Results. The four different models for planetary radio emission lead to very different results. The largest fluxes are found for the magnetic energy model, followed by the CME model and the kinetic energy model (for which our results are found to be much less optimistic than those of previous studies). The unipolar interaction model does not predict any observable emission for the present exoplanet census. We also give estimates for the planetary magnetic dipole moment of all currently known extrasolar planets, which will be useful for other studies.
Conclusions. Our results show that observations of exoplanetary radio emission are feasible, but that the number of promising targets is not very high. The catalog of targets will be particularly useful for current and future radio observation campaigns (e.g. with the VLA, GMRT, UTR-2 and with LOFAR).
Key words: radiation mechanisms: non-thermal - catalogs - plasmas - planets and satellites general
Recent theoretical studies have shown that a large variety of effects have to be considered, e.g. kinetic, magnetic and unipolar interaction between the star (or the stellar wind) and the planet, the influence of the stellar age, the potential role of stellar CMEs, and the influence of different stellar wind models. So far, there is no single publication in which all of these aspects are put together and where the different interaction models are compared extensively. We also discuss the escape of exoplanetary radio emission from its planetary system, which depends on the local stellar wind parameters. As will be shown, this is an additional constraint for detectability, making the emission from several planets impossible to observe.
The first observation attempts go back at least to Yantis et al. (1977). At the beginning, such observations were necessarily unguided ones, as exoplanets had not yet been discovered. Later observation campaigns concentrated on known exoplanetary systems. So far, no detection has been achieved. A list and a comparison of past observation attempts can be found elsewhere (Grießmeier et al. 2006a). Concerning ongoing and future observations, studies are performed or planned at the VLA (Lazio et al. 2004), GMRT (Majid et al. 2006; Winterhalter et al. 2006), UTR2 (Ryabov et al. 2004), and at LOFAR (Farrell et al. 2004). To support these observations and increase their efficiency, it is important to identify the most promising targets.
The target selection for radio observations is based on theoretical estimates which aim at the prediction of the main characteristics of the exoplanetary radio emission. The two most important characteristics are the maximum frequency of the emission and the expected radio flux. The first predictive studies (e.g. Zarka et al. 1997; Farrell et al. 1999) concentrated on only a few exoplanets. A first catalog containing estimations for radio emission of a large number of exoplanets was presented by Lazio et al. (2004). This catalog included 118 planets (i.e. those known as of 2003, July 1) and considered radio emission energised by the kinetic energy of the stellar wind (i.e. the kinetic model, see below). Here, we present a much larger list of targets (i.e. 197 exoplanets found by radial velocity and/or transit searches as of 2007, January 13, taken from http://exoplanet.eu/), and compare the results obtained by all four currently existing interaction models, not all of which were known at the time of the previous overview study. As a byproduct of the radio flux calculation, we obtain estimates for the planetary magnetic dipole moment of all currently known extrasolar planets. These values will be useful for other studies as, e.g., star-planet interaction or atmospheric shielding.
To demonstrate which stellar and planetary parameters are required for the estimation of exoplanetary radio emission, some theoretical results are briefly reviewed (Sect. 2). Then, the sources for the different parameters (and their default values for the case where no measurements are available) are presented (Sect. 3). In Sect. 4, we present our estimations for exoplanetary radio emission. This section also includes estimates for planetary magnetic dipole moments. Section 5 closes with a few concluding remarks.
In principle, there are four different types of interaction between a planetary obstacle and the ambient stellar wind, as both the stellar wind and the planet can either be magnetised or unmagnetised. Zarka (2007, Table 1) show that for three of these four possible situations intense nonthermal radio emission is possible. Only in the case of an unmagnetised stellar wind interacting with an unmagnetised body no intense radio emission is possible.
In those cases where strong emission is possible, the expected radio flux depends on the source of available energy. In the last years, four different energy sources were suggested: a) in the first model, the input power into the magnetosphere is assumed to be proportional to the total kinetic energy flux of the solar wind protons impacting on the magnetopause (Zarka et al. 1997; Farrell et al. 1999; Grießmeier et al. 2005; Farrell et al. 2004; Grießmeier et al. 2006b,2007; Lazio et al. 2004; Zarka et al. 2001b; Stevens 2005; Desch & Kaiser 1984); b) similarly, the input power into the magnetosphere can be assumed to be proportional to the magnetic energy flux or electromagnetic Poynting flux of the interplanetary magnetic field (Farrell et al. 2004; Zarka 2006,2007,2004; Zarka et al. 2001b). From the data obtained in the solar system, it is not possible to distinguish which of these models is more appropriate (the constants of proportionality implied in the relations given below are not well known, see Zarka et al. 2001b), so that both models have to be considered; c) for unmagnetised or weakly magnetised planets, one may apply the unipolar interaction model. In this model, the star-planet system can be seen as a giant analog to the Jupiter-Io system (Zarka 2006,2007,2004; Zarka et al. 2001b). Technically, this model is very similar to the magnetic energy model, but the source location is very different: whereas in the kinetic and in the magnetic model, the emission is generated near the planet, in the unipolar interaction case a large-scale current system is generated and the radio emission is generated in the stellar wind between the star and the planet. Thus, the emission can originate from a location close to the stellar surface, close to the planetary surface, or at any point between the two. This is possible in those cases where the solar wind speed is lower than the Alfvén velocity (i.e. for close-in planets, see e.g. Preusse et al. 2005). Previous studied have indicated that this emission is unlikely to be detectable, except for stars with an extremely strong magnetic field (Zarka 2006,2007,2004; Zarka et al. 2001b). Nevertheless, we will check whether this type of emission is possible for the known exoplanets; d) the fourth possible energy source is based on the fact that close-in exoplanets are expected to be subject to frequent and violent stellar eruptions (Khodachenko et al. 2007b) similar to solar coronal mass ejections (CMEs). As a variant to the kinetic energy model, the CME model assumes that the energy for the most intense planetary radio emission is provided by CMEs. During periods of such CME-driven radio activity, considerably higher radio flux levels can be achieved than during quiet stellar conditions (Grießmeier et al. 2006b,2007). For this reason, this model is treated separately.
For the kinetic energy case, the input power was first derived by
Desch & Kaiser (1984), who found that it is given by
The magnetic energy case was first discussed by Zarka et al. (2001b).
Here, the input power is given by
For the unipolar interaction case (Zarka et al. 2001b), the input power is given by
CME-driven radio emission was first calculated by Grießmeier et al. (2006b). In that case, the input power is given by
A certain fraction
of the input power
given by Eqs. (1), (2), (3) or (4) is thought to be dissipated within the magnetosphere:
The radio flux
seen by an observer at a distance s from the emitter is related to the emitted radio power
by (Grießmeier et al. 2007):
The radio flux expected for the four different models according to Eqs. (1) to (7) and the maximum emission frequency according to (8) are calculated in Sect. 4 for all known exoplanets.
To allow an observation of exoplanetary radio emission, it is not sufficient to have a high enough emission power at the source and emission in an observable frequency range. As an additional requirement, it has to be checked that the emission can propagate from the source to the observer. This is not the case if the emission is absorbed or trapped (e.g. in the stellar wind
in the vicinity of the radio-source),
which happens whenever the plasma frequency
Note however that, depending on the line of sight, not all observers will be able to see the planetary emission at all times. For example, the observation of a secondary transit implies that the line of sight passes very close to the planetary host star, where the plasma density is much higher. For this reason, some parts of the orbit may be unobservable even for planets for which Eq. (10) is satisfied.
An additional constraint arises because certain conditions are necessary for the generation of radio emission.
Planetary radio emission is caused by the cyclotron maser instability (CMI). This mechanism is only efficient
in regions where the ratio between the electron plasma frequency and the electron cyclotron frequency is small enough. This condition can be written as
The condition imposed by Eq. (11) has to be fulfilled for any of the four models presented in Sect. 2.1. For the three models where the radio emission is generated directly in the planetary magnetosphere, n decreases much faster with distance to the planetary surface than B, so that Eq. (11) can always be fulfilled. For the unipolar interaction model, the emission takes place in the stellar wind, and the electron density n can be obtained from the model of the stellar wind. In this case, it is not a priori clear whether and where radio emission is possible. It could be generated anywhere between the star and the planet. In Sect. 4, we will check separately for each planetary system whether unipolar interaction satisfying Eq. (11) is possible at any location between the stellar surface and the planetary orbit.
In the previous section, it has been shown that the detectability of planetary radio emission depends on a few planetary parameters:
As a first step, basic planetary characteristics have to be evaluated:
The stellar wind density n and velocity encountered by a planet are key parameters defining the size of the magnetosphere and thus the energy flux available to create planetary radio emission. As these stellar wind parameters strongly depend on the stellar age, the expected radio flux is a function of the estimated age of the exoplanetary host star (Stevens 2005; Grießmeier et al. 2005). At the same time it is known that at close distances the stellar wind velocity has not yet reached the value it has at larger orbital distances. For this reason, a distance-dependent stellar wind models has to be used to avoid overestimating the expected planetary radio flux (Grießmeier 2006; Grießmeier et al. 2007).
It was shown (Grießmeier et al. 2007) that for stellar ages >0.7 Gyr, the radial dependence of the stellar wind properties can be described by the stellar wind model of Parker (1958), and that the more complex model of Weber & Davis (1967) is not required. In the Parker model, the interplay between stellar gravitation and pressure gradients leads to a supersonic gas flow for sufficiently large substellar distances d. The free parameters are the coronal temperature and the stellar mass loss. They are indirectly chosen by setting the stellar wind conditions at 1 AU. More details on the model can be found elsewhere (e.g. Mann et al. 1999; Preusse 2006; Grießmeier 2006).
The dependence of the stellar wind density n and velocity and
on the age of the stellar system is based on observations of astrospheric absorption features of stars with different ages.
In the region between the astropause and the astrospheric bow shock (analogs to the heliopause
and the heliospheric bow shock of the solar system), the partially ionized local
interstellar medium (LISM) is heated and compressed. Through charge exchange processes,
a population of neutral hydrogen atoms with high temperature is created. The characteristic
absorption (at 1216 )
of this population was detectable with the high-resolution
observations obtained by the Hubble Space Telescope (HST).
The amount of absorption depends on the size of the astrosphere, which is a function of the
stellar wind characteristics.
Comparing the measured absorption to that calculated by hydrodynamic codes, these measurements
allowed the first empirical estimation of the evolution of the stellar mass loss rate as a
function of stellar age (Wood 2004; Wood et al. 2005,2002). It should be noted, however, that the resulting estimates are only valid for stellar ages 0.7 Gyr (Wood et al. 2005).
From these observations, (Wood et al. 2005) calculate the age-dependent density of the stellar wind under the assumption of an age-independent stellar wind velocity.
This leads to strongly overestimated stellar wind densities, especially for young stars (Grießmeier et al. 2005; Holzwarth & Jardine 2007). For this reason, we combine these results with the model for the
age-dependence of the stellar wind velocity of Newkirk (1980). One obtains
(Grießmeier et al. 2007):
For planets at small orbital distances, the keplerian velocity of the planet moving around its
star becomes comparable to the radial stellar wind velocity. Thus,
the interaction of the stellar wind with the planetary magnetosphere should be calculated using
the effective velocity of the stellar wind plasma relative to the planet, which takes into
account this "aberration effect''
(Zarka et al. 2001b).
For the small orbital distances relevant for Hot Jupiters, the planetary orbits are circular
because of tidal dissipation (Halbwachs et al. 2005; Goldreich & Soter 1966; Dobbs-Dixon et al. 2004).
For circular orbits, the orbital velocity
is perpendicular to
the stellar wind velocity v, and its value is given by Kepler's law.
In the reference frame of the planet, the stellar wind velocity then is given by
Note that first measurements of stellar magnetic fields for planet-hosting stars are just becoming available from the spectropolarimeter ESPaDOnS (Catala et al. 2007). This will lead to an improved understanding of stellar magnetic fields, making more accurate models possible in the future.
For the CME-driven radio emission, the stellar wind parameters n and are effectively replaced by the corresponding CME parameters and , potentially leading to much more intense radio emission than those driven by the kinetic energy of the steady stellar wind (Grießmeier 2006; Grießmeier et al. 2006b,2007).
These CME parameters are estimated by Khodachenko et al. (2007b), who combine in-situ measurements near the sun (e.g. by Helios) with remote solar observation by SoHO. Two interpolated limiting cases are given, denoted as weak and strong CMEs, respectively. These two classes have a different dependence of the average density on the distance to the star d. In the following, these quantities will be labeled and , respectively.
For weak CMEs, the density
For strong CMEs, Khodachenko et al. (2007b) find
As far as the CME velocity is concerned, one has to note that individual CMEs have very
different velocities. However, the average CME velocity v is approximately independent
of the subsolar distance, and is similar for both types of CMEs:
For each planet, the value of the planetary magnetic moment is estimated by taking the geometrical mean of the maximum and minimum result obtained by different scaling laws. The associated uncertainty was discussed by Grießmeier et al. (2007). The different scaling laws are compared, e.g., by Farrell et al. (1999), Grießmeier et al. (2004) and Grießmeier (2006). In order to be able to apply these scaling laws, some assumptions on the planetary size and structure are required. The variables required in the scaling laws are (the radius of the dynamo region within the planet), (the density within this region), (the conductivity within this region) and (the planetary rotation rate).
profile within the planet
is obtained by describing the planet as a polytropic gas sphere, using the solution of the
Lane-Emden equation (Chandrasekhar 1957; Sánchez-Lavega 2004):
The average density in the dynamo region is then obtained by averaging the density over the range . For Jupiter, we obtain kg m-3.
Depending on the orbital distance of the planet and the timescale for synchronous rotation (which is derived in Appendix B), three cases can be distinguished:
The size of the planetary magnetosphere
is calculated with the parameters determined above for
the stellar wind and the planetary magnetic moment.
For a given planetary orbital distance d, of the different pressure contributions
only the magnetospheric magnetic pressure is a function of
the distance to the planet. Thus, the standoff distance
is found from the pressure
(Grießmeier et al. 2005):
Note that in a few cases, especially for planets with very weak magnetic moments and/or subject to dense and fast stellar winds of young stars, Eq. (30) yields standoff distances , where is the planetary radius. Because the magnetosphere cannot be compressed to sizes smaller than the planetary radius, we set in those cases.
Table 1 shows what radio emission we expect from the presently known exoplanets (13.1.2007). It contains the maximum emission frequency according to Eq. (8), the plasma frequency in the stellar wind at the planetary location according to Eq. (9) and the expected radio flux according to the magnetic model (1), the kinetic model (2), and the kinetic CME model (4). The unipolar interaction model is discussed in the text below. Table 1 also contains values for the expected planetary mass , its radius and its planetary magnetic dipole moment . For each planet, we note whether tidal locking should be expected.
Note that http://exoplanet.eu contains a few more planets than Table 1, because for some planets, essential data required for the radio flux estimation are not available (typically s, the distance to the observer).
The numbers given in Table 1 are not accurate results, but should be regarded as refined estimations intended to guide observations. Still, the errors and uncertainties involved in these estimations can be considerable. As was shown in Grießmeier et al. (2007), the uncertainty on the radio flux at Earth, , is dominated by the uncertainty in the stellar age (for which the error is estimated as 50% by Saffe et al. 2005). For the maximum emission frequency, , the error is determined by the uncertainty in the planetary magnetic moment , which is uncertain by a factor of two. For the planet Bootes b, these effects translate into an uncertainty of almost one order of magnitude for the flux (the error is smaller for planets around stars of solar age), and an uncertainty of a factor of 2-3 for the maximum emission frequency. This error estimate is derived and discussed in more detail by Grießmeier et al. (2007).
The results given in Table 1 cover the following range:
Table 1 also shows that planets subject to tidal locking have a smaller magnetic moment and thus a lower maximum emission frequency than freely rotating planets. The reduced bandwidth of the emission can lead to an increase of the radio flux, but frequently emission is limited to frequencies not observable on earth (i.e. below the ionospheric cutoff).
The results of Table 1 are visualized in Fig. 1 (for the magnetic energy model), Fig. 2 (for the CME model) and Fig. 3 (for the kinetic energy model). The predicted planetary radio emission is denoted by open triangles (two for each "potentially locked'' planet, otherwise one per planet). The typical uncertainties (approx. one order of magnitude for the flux, and a factor of 2-3 for the maximum emission frequency) are indicated by the arrows in the upper right corner. The sensitivity limit of previous observation attempts are shown as filled symbols and as solid lines (a more detailed comparison of these observations can be found in Grießmeier et al. 2006a; Grießmeier 2006; Zarka 2004). The expected sensitivity of new and future detectors (for 1 h integration and 4 MHz bandwidth, or any equivalent combination) is shown for comparison. Dashed line: upgraded UTR-2, dash-dotted lines: low band and high band of LOFAR, left dotted line: LWA, right dotted line: SKA. The instruments' sensitivities are defined by the radio sky background. For a given instrument, a planet is observable if it is located either above the instrument's symbol or above and to its right. Again, large differences in expected flux densities are apparent between the different models. On average, the magnetic energy model yields the largest flux densities, and the kinetic energy model yields the lowest values. Depending on the model, between one and three planets are likely to be observable using the upgraded system of UTR-2. Somewhat higher numbers are found for LOFAR. Considering the uncertainties mentioned above, these numbers should not be taken literally, but should be seen as an indicator that while observation seem feasible, the number of suitable candidates is rather low. It can be seen that the maximum emission frequency of many planets lies below the ionospheric cutoff frequency, making earth-based observation of these planets impossible. A moon-based radio telescope however would give access to radio emission with frequencies of a few MHz (Zarka 2007). As can be seen in Figs. 1-3, this frequency range includes a significant number of potential target planets with relatively high flux densities.
Figures 1-3 also show that the relatively high frequencies of the LOFAR high band and of the SKA telescope are probably not very well suited for the search for exoplanetary radio emission. These instruments could, however, be used to search for radio emission generated by unipolar interaction between planets and strongly magnetised stars.
|Figure 1: Maximum emission frequency and expected radio flux for known extrasolar planets according to the magnetic energy model, compared to the limits of past and planned observation attempts. Open triangles: Predictions for planets. Solid lines and filled circles: previous observation attempts at the UTR-2 (solid lines), at Clark Lake (filled triangle), at the VLA (filled circles), and at the GMRT (filled rectangle). For comparison, the expected sensitivity of new detectors is shown: upgraded UTR-2 (dashed line), LOFAR (dash-dotted lines, one for the low band and one for the high band antenna), LWA (left dotted line) and SKA (right dotted line). Frequencies below 10 MHz are not observable from the ground (ionospheric cutoff). Typical uncertainties are indicated by the arrows in the upper right corner.|
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|Figure 2: Maximum emission frequency and expected radio flux for known extrasolar planets according to the CME model, compared to the limits of past and planned observation attempts. Open triangles: predictions for planets. All other lines and symbols are as defined in Fig. 1.|
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|Figure 3: Maximum emission frequency and expected radio flux for known extrasolar planets according to the kinetic energy model, compared to the limits of past and planned observation attempts. Open triangles: predictions for planets. All other lines and symbols are as defined in Fig. 1.|
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According to our analysis, the best candidates are:
Considering the uncertainties mentioned above, it is important not to limit observations attempts to these best cases. The estimated radio characteristics should only be used as a guide (e.g. for the target selection, or for statistical analysis), but individual results should not be regarded as precise values.
It may seem surprising that so few good candidates are found among the 197 examined exoplanets. However, when one checks the list of criteria for "good'' candidates (e.g. Grießmeier et al. 2006a), it is easily seen that only a few good targets can be expected: (a) the planet should be close to the Earth (otherwise the received flux is too weak). About 70% of the known exoplanets are located within 50 pc, so that this is not a strong restriction; (b) a strongly magnetised system is required (especially to obtain frequencies above the ionospheric cutoff). For this reason, the planet should be massive (as seen above, we find magnetic moments only for planets with masses ). About 60% of the known exoplanets are at least as massive as Jupiter (but only 40% have ); (c) the planet should be located close to its host star to allow for strong interaction (dense stellar wind, strong stellar magnetic field). Only 25% of the known exoplanets are located within 0.1 AU of their host star.
By multiplying these probabilities, one finds that close ( pc), heavy ( ), close-in ( AU) planets would represent 8% (15 of 197 planets) of the current total if the probabilities for the three conditions were independent. However, this is not the case. In the current census of exoplanets, a correlation between planetary mass and orbital distance is clearly evident, with a lack of close-in massive planets (see e.g. Udry et al. 2003). This is not a selection effect, as massive close-in planets should be easier to detect than low-mass planets. This correlation was explained by the stronger tidal interaction effects for massive planets, leading to a faster decrease of the planetary orbital radius until the planet reaches the stellar Roche limit and is effectively destroyed (Jiang et al. 2003; Pätzold & Rauer 2002). Because of this mass-orbit correlation, the fraction of good candidates is somewhat lower (approx. 2%, namely 3 of 197 planets: HD 41004 B b, Tau Boo b, and HD 162020 b).
A first comparative study of expected exoplanetary radio emission from a large number of planets was performed by Lazio et al. (2004), who compared expected radio fluxes of 118 planets (i.e. those known as of 2003, July 1). Their results differ considerably from those given in Table 1:
Predictions concerning the radio emission from all presently known extrasolar planets were presented. The main parameters related to such an emission were analyzed, namely the planetary magnetic dipole moments, the maximum frequency of the radio emission, the radio flux densities, and the possible escape of the radiation towards a remote observer.
We compared the results obtained with various theoretical models. Our results confirm that the four different models for planetary radio emission lead to very different results. As expected, the largest fluxes are found for the magnetic energy model, followed by the CME model and the kinetic energy model. The results obtained by the latter model are found to be less optimistic than by previous studies. The unipolar interaction model does not lead to observable emission for any of the currently known planets. As it is currently not clear which of these models best describes the auroral radio emission, it is not sufficient to restrict oneself to one scaling law (e.g. the one yielding the largest radio flux). Once exoplanetary radio emission is detected, observations will be used to constrain and improve the model.
These results will be particularly useful for the target selection of current and future radio observation campaigns (e.g. with the VLA, GMRT, UTR-2 and with LOFAR). We have shown that observation seem feasible, but that the number of suitable candidates is relatively low. The best candidates appear to be HD 41004 B b, Epsilon Eridani b, Tau Boo b, HD 189733 b, Gliese 876 c, HD 73256 b, and GJ 3021 b. The observation of some of these candidates is in progress.
We thank J. Schneider for providing data via "The extrasolar planet encyclopedia'' (http://exoplanet.eu/), I. Baraffe and C. Vocks for helpful discussions concerning planetary radii. We would also like to thank the anonymous referee for his helpful comments. This study was jointly performed within the ANR project "La détection directe des exoplanètes en ondes radio'' and within the LOFAR transients key project (TKP). J.-M. G. was supported by the french national research agency (ANR) within the project with the contract number NT05-1_42530 and partially by Europlanet (N3 activity). P.Z. acknowledges support from the International Space Science Institute (ISSI) within the ISSI team "Search for Radio Emissions from Extra-Solar Planets''.
For the selection of targets for the search for radio emission from extrasolar planets, an estimation of the expected radio flux and of the maximum emission frequency is required. For the calculation of these values, both the planetary mass and the planetary radius are required (see, e.g. Farrell et al. 1999; Zarka 2007; Grießmeier et al. 2007). However, only for a few planets (i.e. the 16 presently known transiting planets) both mass and radius are known.
In the absence of observational data, it is in principle possible to obtain planetary radii from numerical simulation, e.g. similar to those of Bodenheimer et al. (2003) or Baraffe et al. (2005,2003), requiring one numerical run per planet. We chose instead to derive a simplified analytical fit to such numerical results.
The description presented here is necessarily only preliminary, as (a) numerical models are steadily further developed and improved, and as (b) more transit observations (e.g. by the COROT satellite, which was launched recently) will provide a much better database in the future. This will considerably improve our understanding of the dependence of the planetary radius on various parameters as, e.g. planetary mass, orbital distance, or stellar metallicity (as suggested by Guillot et al. 2006).
Within the frame of the models presented in Sect. 2, an increase in by 40% increases the expected radio flux by a factor of 2, and the estimated maximum emission frequency decreases by 40%. More generally, for a fixed planetary mass, is roughly proportional to and is approximately proportional to . Thus, it appears that the assumption of a single standard radius for all planets leads to a relatively large error. Comparing this to the other uncertainties involved in the estimation of radio characteristics (these are discussed in Grießmeier et al. 2007), it seems sufficient to estimate with 20% accuracy.
Several effects limit the precision in planetary radius we can hope to achieve:
A simple mass-radius relation, valid within a vast mass range, has been proposed by
Lynden-Bell & O'Dwyer (2001) and Lynden-Bell & Tout (2001):
It is known that the radius of a planet with a given mass depends on its age.
According to models for the radii of isolated planets (Baraffe et al. 2003),
the assumption of a time-independent planetary radius leads to an error 11% for planets with ages above 0.5 Gyr and with masses above
In view of the uncertainties discussed above, this error seems acceptable for a first approximation, and we use
It is commonly expected that planets subjected to strong stellar radiation have a larger planetary radius than isolated, but otherwise identical planets.
This situation is typical for "Hot Jupiters'', where strong stellar irradiation is supposed
to delay the planetary contraction
(Burrows et al. 2004,2000,2003).
Clearly, this effect depends on the planetary distance to its star, d, and on the stellar luminosity
In the following, we denote the radius increase by r, which we define as
Different exoplanets have vastly different host stars. A difference of a factor of two in stellar mass can result in a difference of more than order of magnitude in stellar luminosity. For this reason, it is not sufficient to take the orbital distance as the only parameter determining the radius increase by irradiation.
Here, we select the equilibrium temperature of the planetary surface as the basic parameter.
It is defined as
(Bodenheimer et al. 2003)
Using the data of Bodenheimer et al. (2003), we use the following fit for the radius increase r:
The numerical results of Bodenheimer et al. (2003) show that the ratio r also depends on the mass of the planet: for small planets, r=1.4 for an equilibrium temperature of 2000 K, whereas for large planets, .
For this reason, we allow to coefficients T0 and
to vary with
For comparison, Fig. A.1 also shows the value of r for the transiting exoplanets OGLE-TR-10b, OGLE-TR-56b, OGLE-TR-111b, OGLE-TR-113b, OGLE-TR-132b, XO-1b, HD 189733b, HD 209458b, TrES-1b and TrES-2b as small crosses. Here, r is calculated as the ratio of the observed value of and the value for calculated according to Eq. (A.1). As the mass of all transiting planets lies between , one should expect to find all crosses between the two limiting curves. For most planets, this is indeed the case: only for HD 209458b, r is considerably outside the area delimited by the two curves. Different explanations have been put forward for the anomalously large radius of this planet, but so far no conclusive answer to this question has been found (see e.g. Guillot et al. 2006, and references therein). As numerical models cannot reproduce the observed radius of this planet, one cannot expect our approach (which is based on a fit to numerical results) to reproduce it either. For the other planets, Fig. A.1 shows that our approach is a valid approximation.
|Figure A.1: Radius increase r as a function of the equilibrium temperature according to Eqs. (A.5)-(A.7). Upper line: radius increase r due to irradiation for planets of mass . Lower line: radius increase r due to irradiation for planets of mass . Open symbols: results of the numerical calculation of Bodenheimer et al. (2003) for planetary masses (diamonds) and (triangles). Crosses: radius increase for observed transiting planets (see text).|
For the estimation of the planetary magnetic dipole moment in Sect. 3.4, we require the planetary rotation rate. This rotation rate rotation greatly depends on whether the planet can be considered as tidally locked, as freely rotating, or as potentially locked. In this appendix, we discuss how we evaluate the tidal locking timescale , which decides to which of these categories a planet belongs.
The value of the planetary rotation depends on its distance from the central star. For close-in planets, the planetary rotation rate is reduced by tidal dissipation. In this case, tidal interaction gradually slows down the planetary rotation from its initial value until it reaches the final value after tidal locking is completed.
It should be noted that for planets in eccentric orbits, tidal interaction does not lead to the synchronisation of the planetary rotation with the orbital period. Instead, the rotation period also depends on the orbital eccentricity. At the same time, the timescale to reach this equilibrium rotation rate is reduced (Laskar & Correia 2004). Similarly, for planets in an oblique orbit, the equilibrium rotation period is modified (Levrard et al. 2007). However, the locking of a hot Jupiter in a non-synchronous spin-orbit resonance appears to be unlikely for distances 0.1 AU (Levrard et al. 2007). For this reason, we will calculate the timescale for tidal locking under the assumptions of circular orbits and zero obliquity. In the following, the tidal locking timescale for reaching is calculated under the following simplifying assumptions: prograde orbit, spin parallel to orbit (i.e. zero obliquity), and zero eccentricity (Murray & Dermott 1999, Chap. 4).
The rate of change of the planetary rotation velocity
for a planet with a mass of
and radius of
around a star of mass
is given by (Goldreich & Soter 1966; Murray & Dermott 1999):
The time scale for tidal locking is obtained by a comparison of the planetary angular velocity
and its rate of change:
In the following, we briefly describe the parameters (, , and ) required to calculate the timescale for tidal locking of Hot Jupiters.
For large gaseous planets, the equation of state can be approximated by a polytrope of index . In that case, the structure parameter (defined by ) is given by (Gu et al. 2003).
For planets with masses of the order of one Jupiter mass, one finds that has a value of for Jupiter (Laskar & Correia 2004; Murray & Dermott 1999) and for Saturn (Laskar & Correia 2004; Peale 1999). The value of will be used in this work. With Eq. (B.2), this results in .
For Jupiter, one finds the following range of allowed values: (Peale 1999). Several estimations of the turbulent dissipation within Jupiter yield -values larger than this upper limit, while other theories predict values consistent with this upper limit (Marcy et al. 1997; Peale 1999, and references therein). This demonstrates that the origin of the value of is not well understood even for Jupiter (Marcy et al. 1997).
Extrasolar giant planets are subject to strongly different conditions, and it is difficult to constrain . Typically, Hot Jupiters are assumed to behave similarly to Jupiter, and values in the range are used. In the following, we will distinguish three different regimes: close planets (which are tidally locked), distant planets (which are freely rotating), and planets at intermediate distances (which are potentially tidally locked). The borders between the "tidally locked'' and the "potentially locked'' regime is calculated by setting Myr and . The border between the "potentially locked'' and the "freely rotating'' regime is calculated by setting Gyr and . Thus, the area of "potentially locked'' planets is increased.
The initial rotation rate is not well constrained by planetary formation theories. The relation between the planetary angular momentum density and planetary mass observed in the solar system (MacDonald 1964) suggests a primordial rotation period of the order of 10 h (Hubbard 1984, Chap. 4). We assume the initial rotation rate to be equal to the current rotation rate of Jupiter (i.e. ) with s-1.
As far as Eq. (B.4) is concerned,
can be neglected