### Hyper-Spinning Regulus

May 4, 2011

It's no surprise to anyone with a mechanical sensibility that a spinning orb of elastic matter will bulge at its equator. This is true even when the material is not that elastic, like the rocks that comprise the Earth's crust, so the Earth has a bulge at its equator. The Earth differs somewhat from a sphere; it's an oblate spheroid with a 42.72 km (26.5 mile) bulge at the equator. The oblateness of the Earth is given as
Oblateness = (a - b)/a = 0.335%

where a is the distance from the equatorial surface to the Earth's center, and b is the distance from the Earth's pole to its center.

The gas giant, Saturn, has an oblateness of 9.8%. The Sun's oblateness is only 9 x 10-6, or 0.0009%, which is a consequence of the large gravitational force holding it together, and its slow equatorial rotation rate of 25 days. Contrast this with the star, Regulus, which has an estimated rotation rate of about sixteen hours, and an oblateness of about 26%.[1]

Regulus is known also as α Leonis. It's the brightest star in the constellation Leo, as its "alpha" designation indicates. It's also the twenty-second brightest star (magnitude 1.35) in Earth's sky. When you consider all the stars visible with the naked eye, twenty-second place is quite a high rank. Regulus is about 77.5 light years from Earth.

An image of the star Regulus showing its equatorial bulge.

(Image by Xiao Che, University of Michigan)

To say that Regulus is rapidly-spinning is an understatement. Whereas the Sun's equatorial speed is about 4,500 miles per hour, the equator of Regulus, which has a five-fold larger radius than the Sun and a much faster rotation rate, is 700,000 miles per hour. This has more of an impact if we rephrase this number as nearly a million miles per hour. Seven hundred thousands miles per hour means a trip from Earth to the Moon in just twenty minutes. Calculations show that Regulus would fly apart if its rotation rate were just 16% faster.

Rapid spinning has a big effect on stellar hydrodynamics. The poles of Regulus have a larger surface gravity, and they are hotter and brighter than the equator. The magnitude of this so-called "gravity darkening" of the equator is usually estimated by the von Zeipel Theorem,
T ∼ g1/4

in which the temperature T is proportional to the quarter-power of the local gravity g. The power is only 0.25 if the local energy transfer is purely radiative. It's reduced when convection is present.[2]

Recently, some astronomers from the University of Michigan have used interferometry to do precise temperature measurements of Regulus.[3-4] A star's temperature is an important predictor of its age. They combined the infrared light of four telescopes at the CHARA array of Georgia State University to give a huge effective aperture instrument they could use to examine particular regions of the star.

They found that the temperature difference between equator and poles is considerably less than the von Zeipel Theorem prediction, and it's more consistent with an exponent of 0.19. This may seem like a small difference, but the effect is amplified in Regulus by the very large difference in effective gravity between the equator and poles. The difference between the von Zeipel prediction and observation is about 2750°C.[4]

The asteroid 163 Erigone is expected to pass between Regulus and the Earth on March 20, 2014, a so-called occultation. Occultations were previously the only way to study the fine structure of the asteroids. Now we have spacecraft. One interesting thing about this occultation is that its path is predicted to pass over my house. The occultation path was calculated by Aldo Vitagliano,[5] and his prediction appears in the figure.

Predicted occultation path of Regulus by asteroid 163 Erigone on March 20, 2014.

(Image by Aldo Vitagliano, Ref. 5, Modified.)

### References:

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