Changing views on the radius inflation of hot Jupiters


The “anomalous radius problem”

The first transiting hot Jupiter, HD 209458 b, is much larger than can be accounted for by standard models of structure and evolution of gas giant planets. It has a radius nearly 30% larger than Jupiter. Since it orbits very close to its host star and has an equilibrium temperature above 1500 K, models did predict a radius larger than Jupiter, but only by about 10%.

This is the source of the “anomalous radius problem” of hot Jupiters (also called inflated, or sometimes bloated, hot Jupiters).

In the decade elapsed since the discovery of ‘209, many other inflated hot Jupiters have been identified. Radius inflation seem to affect about half of the close-in gas giant planets, and can sometimes reach extreme values, close to twice the size of Jupiter.

Remember that for giant planets, it is very difficult to exceed the size of Jupiter. The fundamental reason is that the interior of giant planets is mostly in the form of hydrogen and helium supported by pressure from electron degeneracy. In this form, the equation of state is such that the size of the body remains almost constant with increasing mass. For this reason, models predict radii close to Jupiter all the way from lighter Saturn-mass planets to heavy brown dwarfs. The degeneracy is only broken for stars by the heat from nuclear reactions in their core, otherwise they too would be Jupiter-sized.

Mass - radius relation of planetary and stellar bodies. From the website of the Astrophysics Spectator.

Inflating a hot Jupiter from 1.1 to 1.3 Jupiter radii requires a massive injection of energy in the interior of the planet. The irradiation from the star is the first suspect of course, since hot Jupiters differ from our Jupiter mainly by their closeness to their host star and consequent high amount of incoming starlight. But in fact this does not work well to keep the planet large: the star’s light is absorbed in the outermost layers of the planet, heating its atmosphere. As a result, the planet radiates more thermal infrared, quickly establishing a balance with the incoming irradiation, that leaves the interior of the planet unaffected.

Mass-radius diagram for gas giant planets. About half the known hot Jupiters are larger than predicted by any model. The solid blue line labelled "pure H/He" shows the predicted relation for normal hot Jupiters. Dotted lines show the relation for hot Jupiters slightly closer or more distant from the star. The lower solid line shows the relation for planets with a core of heavier elements amounting to 50% of the planet's mass. (Models from Jonathan Fortney).

Wind dissipation?

In the first structure and evolution models of ‘209, Showman & Guillot (2002), proposed a simple and elegant solution to the radius inflation problem. In hot Jupiters, the main effect of stellar irradiation, at least in the numerical simulations, is to establish strong eastwards jets in the atmosphere that brings the heat from the day side to the night side of the planet. If a small fraction, of the order of one percent, of the stellar irradiation energy was somehow converted deep enough into the interior of the planet,  via the dissipation of the atmospheric winds, then the size of ‘209 could be explained.

How deep is “deep enough” though?

This requires a slightly more technical aside into what it means to inflate a hot Jupiter. Actually, speaking of radius inflation is slightly misleading, since this is not what actually happens. Gas giant planets do not generally “inflate”, they are born extremely large, and shrink with time as they loose the initial thermal energy stored in their hot interior after the violent formation stage. Anomalous sizes are most naturally reached not by re-inflating a Jupiter-size planes, but by delaying its contraction.

Now, there is a trick: according to 1-D simulations and solid analytical arguments, the interior of hot Jupiters cools in a slightly peculiar way. Because they loose heat on their surface and convection sets the slope of the interior profile, the way they contract is by a decrease in temperature throughout their interior along the adiabat in the convective zone, until an outer isothermal layer is reached. The temperature in this outer layer is set by the equilibrium with the stellar irradiation. This may be clearer in the figure below.

Evolution of the temperature profile of a hot Jupiter

Therefore, hot Jupiters are expected to evolve with a cooling convective interior, surrounded by an isothermal radiative zone that deepens with time.

The only way to retain an anomalous radius for such a theoretical planet after billions of years is to slow down the deepening of the radiative zone. This can be done either by injecting a large amount of energy into the planet, or by slowing down the cooling. The first case correspond to scenarios involving injection of stellar irradiation energy or orbital energy into the planet; the second to scenarios invoking less efficient heat loss,  either via increased opacities or a flatter temperature profile from reduced convection.

The challenge for the first type of scenarios is to inject the energy deep enough. If the energy is simply used to increase the temperature of the isothermal zone, then it will result in a higher thermal radiation from the planet into space, which would directly compensate the energy injection. To affect the planetary radius over the long term, the energy injection much be felt by the adiabatic zone. According to structure models, this level is between a few tens and a few hundreds of bars for inflated hot Jupiters, in a region not directly affected by the host star irradiation.

It is worth remembering, though, that this radiative-convective structure of hot Jupiters has not been established empirically, and may turn out to be oversimplified. Work is being done to model the interior evolution of gas giant exoplanets with more details.

Tidal dissipation?

An alternative to the wind-dissipation scenario was proposed by Winn & Holman (2005). If the orbit of ‘209 was perturbed by an undetected second planet, it could be excited into a non-circular state, a so-called “Cassini state” after the orbital dynamics of the satellites of Jupiter. The resulting tidal interactions with the star could inject enough energy into the planet to explain the radius anomaly. However, Fabrycky et al. (2007) later showed that this mechanism would not work for ‘209.

As more anomalously large hot Jupiters were discovered (such as the alarming TrES-4 b, with R=1.8 RJup!), and more detailed simulations of atmospheric circulation did not seem to show a natural way to realise the wind-dissipation scenario, other explanations were proposed.

  • When the orbit of a close-in gas giant is not circular, tidal interaction with the star inject very large amounts of energy in the planet, and this option was explored again (Jackson et al. 2008, Miller et al. 2009). Even if the present orbit is circular, past tidal evolution towards circularity could have injected enough energy.
  • Burrows et al. (2007) suggested that if the opacity of the outer layer of the planet were artificially increased, to simulate an over-abundance of heavy elements in the composition of the planet, then the radius would increase. The higher opacity would hinder the cooling of the planet after its formation, keeping it large for longer. The problem, however, is that the overabundance must not extend to the interior of the planet, otherwise its higher density would make it shrink and would counterbalance the opacity effect.
  • Chabrier & Baraffe (2007) proposed that convection in the interior of the planet could be less efficient than in the models. This can happen in the so-called “double-diffusive convection”, a well-known phenomenon were density gradients hinder normal convection. In hot Jupiters, density gradients in the interior could be caused by the sedimentation of heavier elements.

As the number of known hot Jupiters reached fifty or so (around 2009), it became clear that there was a tight relation between radius inflation and the proximity of the parent star. This spoke against the opacity and convection scenarios, at least as an ensemble explanation.

Tidal evolution became the leading explanation and primary focus of the work on the issue. An undetected planetary companion, a slightly eccentric orbit, and a circularisation in the past, all could explain specific cases (see for instance WASP-12).

However, globally, the tidal explanation required many coincidences and fine-tuning. Several new results made it very unlikely as a general explanation for radius inflation:

  • first, it turned out that many measurements of orbital eccentricity used to calculate the tidal dissipation were spurious, actual circular orbits were thought to be eccentric because of instrumental uncertainties and statistical biases (Husnoo et al. 2011).
  • second, none of the expected perturbing companions were detected, and in some cases, strict exclusion zones could be set, notably by using the lack of variations in the transit timings.
  • third, new calculations of tidal evolution showed that past circularisation produced only a short-lived state of radius inflation (Leconte et al. 2010), that could explain one or two specific case, but not the observed fact that a large fraction of hot Jupiters were inflated.
  • finally, with even more known transiting gas giants, it appears that the primary dependence of radius inflation is with incoming stellar irradiation, not with closeness to the parent star. Of course the two are related, but thanks to the steepness of the mass-luminosity relation of stars, with enough cases it is possible to distinguish between the two dependences. Tidal interactions would yield a mass dependence rather than a luminosity dependence.

Magnetic dissipation?

In 2010, a new type of explanation was proposed, that seems to tie in all the elements very nicely. The atmospheres of hot Jupiters are so hot that some atoms are ionised (mainly alkali metals), and if the planet possesses a magnetic field, the motion of charged particles along the strong atmospheric jets would give rise to Ohmic dissipation. Order-of-magnitude calculations showed that this mechanism could provide the needed energy injection in the planet.

The observed correlation between radius inflation and stellar irradiation also supports the initial wind-dissipation scenario. However, more detailed simulations do no seem to show any significant dissipation of the winds deep enough in the atmosphere (Burkert et al. 2005). Less direct links between winds and dissipation have been proposed, such as large-scale mixing (Youdin & Mitchell 2010) or the dissipation of supersonic shocks (Perna et al. 2012).

So this is were we are at the moment. Although a large part of the community has now swung behind the Ohmic dissipation hypothesis, I think the jury is still very much out on the matter. The magnetic scenario has enough fiddle factors that it is very difficult to test empirically. In particular, it depends strongly on the strength of the magnetic field in hot Jupiters, which is totally unknown.

Nevertheless, the link between atmospheric circulation and radius inflation is now firmly established, and I think we can say with reasonable confidence that the mechanism responsible is the dissipation of the energy of day-night recirculation through some non-trivial mechanism, be it electromagnetic or turbulent-flow dissipation.

Changing views on the radius inflation of hot Jupiters

Feature Image: Mass-radius plot for gas giant planets, data from our transiting planet database, models from Jonathan Fortney.


About Author

I am a professor of planetary science at the University of Exeter. My specialty is the study of exoplanets, in particular the observation and modelling of exoplanet atmospheres. I have done my PhD a the University of Geneva and worked in Chile, France and Switzerland.