Brown dwarfs (BDs) are neither planets nor stars but they share some properties of both. In terms of masses (MBD) they settle an intermediate regime, with 10MJup≲MBD≲75MJup≈0.07MSun, where MJup is the mass of the giant planet Jupiter and MSun is the mass of the Sun. When they are young, BDs can be as hot as cool stars with surface temperatures of ≈2500K. There are several models for the formation scenarios of BDs and for their evolution and structure. In order to discriminate between theses theories, measurements of masses, radii and luminosities of such objects are crucial.
Today, there is only one known BD eclipsing binary, i.e. a system with two BD components that occult each other twice per orbit for an observer on Earth. The cryptic name of this system is 2MASSJ05352184−0546085 (2M0535−05). During each orbital cycle there is one primary and one secondary transit. These events allow for a very precise measurement of the objects' radii in combination with their relative temperatures. Surprisingly, in 2M0535−05 the primary component, i.e. the more massive one, is cooler (surface or effective temperature Teff≈2700K) than the secondary (Teff≈2815K) . This can not be explained by any of the available BD theories. Together with Rory Barnes, Brian Jackson and Richard Greenberg, all of which were employed at the LPL in Tucson (Arizona) those days, I explored tidal heating in this interesting BD binary. Our goal was to find out if it could be responsible for the observed temperature reversal.
Both BDs are moving around their common center of mass on an eccentric orbit. When they come very close to each other they are stretched and squeezed but the deformation slackens more or less as the bodies depart from each other. The distorted shapes of the BDs are thus rather ellipsoids than balls. The rotational periods of both BDs in 2M0535−05 are not synchronous with the orbital period. Thus, the tidal bulges on the BDs yield a rotational acceleration for the one that is rotating more slowly and a deceleration for the BD that turning faster than the orbital period. These processes and the conservation of angular momentum within the two-body system result in a tidal heating of both constituents and an evolution of the orbital parameters.
Below, you find four figures from our paper . You see the tidal energy rates Ėtid (following the tidal model of ) released within the BDs in the left column. The upper row belongs to the primary, the lower one to the secondary. The images show a projection of the heating rates onto the plane spanned by the quality factor Q1 of the primary and the obliquity of the primary ψ1. While Q1 is a measure for the response of the primary to the tidal deformation, ψ1 is a potential, but unknown, misalignment between the primary's equatorial plane and the orbital plane. As you can see: for a fixed Q1, an increasing obliquity also increases the tidal heating. In the right column I plotted the resulting temperature increase assuming that half of the energy rate is converted into a surface temperature increase, while the other half would go into inflation of the BD's radius. Again, the lower Q1 and the higher the obliquity ψ the higher the tidal heating Ėtid.
For Q2=103.5 and ψ2=50° we find a temperature increase dT2 on the secondary's surface of ≈60K, which is about half the temperature excess that is observed. In the same Q-ψ regime for the primary, we get only dT1≈40K. We thus find a reversal in Teff increase, analog to the observed Teff reversal. As mentioned in the literature [1, 4], the large spot coverage on the primary probably decreases its temperature, so that the Teff increase due to tides would be more than compensated. Thus, while intense spot coverage on the primary counterbalances tidal heating, the seconday might be heated significantly.
But is it reasonable to assume such low Q values for BDs and such a high obliquity of the secondary? The Q values of the giant planets and M dwarfs are typically of order 105. If Q1 and Q2 would really be as small as 103.5, then the synchronization time of the system, i.e. the time needed to synchronize orbital and rotational periods, would be of order 0.1 Myr. However, the system is obviously not synchronized although it has an age of 1 Myr. This argument makes such low Q values for BDs very unlikely. But as can be seen in the plots above, tidal heating incewases for a fixed Q when ψ increases. So how about tidal obliquity heating? Is the system young enough that there might exist a significant spin-orbit misalignment of the secondary BD such that is heated by tidal friction? Or would any initial obliquity already have been washed out over the lifetime of the system?
In the figures below, you see two plots of the system's orbital evolution. At the left, there is the evolution of the eccentricity e. On the abscissa you go back in time (from right to left) until you reach −1.5 Million years (Myr) at the left end. The various dotted lines correspond to different values for Q. For this calculation, we assumed that Q1=Q2=Q, i.e. that the two BDs have similar tidal responses. The left figure shows the backwards evolution of e in 2M0535−05 within the last 1.5 Myr: The further one goes back in time and the lower Q again, the more significant is the change in e. But anyhow, this simulation shows that this pretty young system has not evolved very far in terms of orbital properties. In the right panel, you see the backwards evolution of ψ2 within the recent 1.5 Myr. You see that for any of the tested Q2 values ψ2 only slightly increases going backwards in time. We conclude that any initial obliquity still persists today.
We conclude that the tidal quality factors of BDs are higher than ≈103.5, which comes from our considerations of tidal equilibrium rotation not yet achieved in the 2M0535−05 BD binary system. Moreover, tidal heating could be the source of the observed temperature reversal if the obliquity of the secondary mass BD is ≳50°. And the system is young enough that any initial obliquity would still exist today.
 Stassun et al., 2007, ApJ, 664, 1154 [ADS]
 Heller et al., 2010, A&A, 514, 22 [ADS]
 Wisdom, J., 2008, Icarus, 193, 637 [ADS]
 Chabrier et al., 2007, A&A, 472, L17 [ADS]