Main sequence stars of spectral types earlier than ~ F6 (i.e. stars > 1.5 times the Sun’s mass) are expected to be rapid rotators as their outermost layers are radiative instead of convective. Stars like the Sun have convective outermost layers that drive surface magnetic activity, generating strong stellar winds that with time carry away the star’s angular momentum and spin down the star. Stars of spectral types earlier than ~ F6 retain their high angular momentum (i.e. rapid rotation rates) since their radiative outermost layers do not produce strong stellar winds that sap the star’s angular momentum with time. Some of these stars are known to spin at near their break-up speeds.
The rapid rotation of ~ F6 type and earlier stars can make then significantly oblate - flattened at the poles and bulging around the equator. This causes the star’s surface gravity to exhibit a pole-to-equator gradient, with the poles having a higher surface gravity, and thus higher temperature and brightness. The star’s photosphere can be several thousand Kevin hotter at the poles than at the equator. As a result, the poles are “gravity-brightened” and the equator “gravity-darkened”. This effect is called gravity darkening and it was first predicted by von Zeipel (1924).
Figure 1: Artist’s impression of a gas giant planet. Image credit: Daniel Mallia.
A study by Ahlers et al. (2014) used precise photometric data from NASA’s Kepler space telescope to constrain the spin-orbit alignment of the KOI-368 planetary system. Basically, the spin-orbit alignment is the inclination of a planet’s orbit normal with respect to its star’s spin axis. The planet in this system, designated KOI-368.01, is probably an inflated gas giant 1.83 times the size of Jupiter. The planet’s host star, KOI-368, is a rapidly rotating early spectral type star that is 2.3 times the size of the Sun and has an effective temperature of 9257 K. The star’s period of rotation is a mere 30.73 hours. For comparison, the Sun has a rotation period of 25 days.
Gravity darkening allows the true orientation of a star’s spin axis to be determined. In turn, this allows the true spin-orbit alignment of any planet transiting the star to be measured. The planet KOI-368.01 transits its host star every 110 days. Due to the rapid spin of its host star, the transit light curve is expected to display the effects of gravity darkening. Using a gravity-darkened stellar model to model the observed light curve when KOI-368.01 transits its host star, the planet’s true spin-orbit alignment was found to be 11 ± 3 degrees. The results from this study show that the orbit of KOI-368.01 is well aligned and might suggest that orbits of objects circling more massive stars are not more likely to be misaligned, contrary to Winn et al. (2010).
Figure 2: Photometry and fits for the KOI-368 light curve with the gravity-darkened fit in blue. Source: Ahlers et al. (2014).
Figure 3: Four possible transit geometries of the KOI-368 system. The effects of gravity darkening and limb darkening are illustrated. All four scenarios produce identical transit light curves; therefore, these geometries are perfectly degenerate. Source: Ahlers et al. (2014).
- Ahlers et al. (2014), “Spin-Orbit Alignment for 110 Day Period KOI368.01 from Gravity Darkening”, Astrophysical Journal, Volume 786:131 (5pp)
- von Zeipel (1924), “The radiative equilibrium of a rotating system of gaseous masses”, Monthly Notices of the Royal Astronomical Society, Vol. 84, p.665-683
- Winn et al. (2010), “Hot Stars with Hot Jupiters Have High Obliquities”, Astrophysical Journal Letters, Volume 718, Issue 2, pp. L145-L149