5.1 The oblateness controversy

Interest in the rotation of the deep solar interior predates systematic helioseismic observation. One other possible diagnostic of the internal rotation is provided by the solar oblateness; because the Sun is not a solid body, both gravitational and rotational effects cause a very slight flattening. The lowest-order term in this effect is related to the quadrupole moment J2; confusingly, the next-highest term, J4, is sometimes called octopole and sometimes hexadecapole. According to Rozelot and Rösch (1997), who give a useful review of attempts to measure the solar oblateness, for a non-rotating Sun the oblateness Δr = req − rpol is given by
Δr- 3- r = 2J2, (24 ) 0
where req and rpol are the equatorial and polar radii, respectively, and r0 is the radius of the best sphere passing through req and rpol. If there is an additional δr contribution from the surface rotation this expression becomes
Δr − δr 3 -------- = -J2. (25 ) r0 2
The units of δr and Δr are conventionally arc ms.

Dicke (1964Jump To The Next Citation Point) noted that, if the Sun were oblate because of fast interior rotation, the effect on its gravitational potential might destroy the agreement between the predictions of General Relativity and the observations of the perihelia of the inner planets, (specifically Mercury, though Venus could in principle experience a smaller effect) potentially leaving room for alternate theories of gravitation. Dicke sets out to determine the solar oblateness from ground-based measurements – a challenging endeavor that produced controversial results. Models (e.g., Brandt 1966) suggested that the interior of the Sun could still be spinning at the rapid rate at which it originally formed, while the exterior had been slowed down by the torque of the solar wind. (As will be further discussed in Section 6, in the absence of direct observations of the solar interior the picture of solar interior dynamics was not at all clear, although the existence of something like what we now call the tachocline could be inferred from theoretical arguments.) Dicke and Goldenberg (1967bJump To The Next Citation Point) reported finding a solar oblateness value of 5 × 10–5, which would be sufficient to create an 8% discrepancy between observations and the Einsteinian prediction for the precession of the perihelion of Mercury, and would imply a fast-rotating core.

The results, and the inferences Dicke and collaborators drew from them, raised a storm of controversy that may well have helped to stimulate interest in the Sun’s interior rotation profile. The criticisms and Dicke’s responses to them would fill a lengthy review article by themselves; we give only a few examples here. Roxburgh (1967) suggested that the result might be explained by the solar differential rotation, an idea rejected by Dicke and Goldenberg (1967a). Howard et al. (1967) concluded, on the basis of a variety of simple models of the solar “spin-down,” that the Sun should have reached a state of uniform rotation quite quickly after its initial formation. Sturrock and Gilvarry (1967) pointed out that the presence of magnetic field in the solar interior might well complicate the issue, and in an accompanying article Gilvarry and Sturrock (1967) suggested using a space probe in a highly eccentric orbit as a more direct test of general relativity – or, alternatively, that “more complete theoretical and observational knowledge of the visible layers and the interior of the Sun” was needed.

At least partly inspired by the controversy, Kraft (1967) studied the rotational velocities of young solar-type stars in the Pleiades and concluded that angular momentum was lost on a timescale of about half a billion years, but noted in his conclusion that “it is wrong to conclude that the present work in any way supports the Dicke result.” Goldreich and Schubert (1968) considered the stability of differentially rotating stars and concluded that it was possible but not likely that a radial rotation gradient such as that required by the Dicke and Goldenberg (1967bJump To The Next Citation Point) result might exist.

H. Hill, a former colleague of Dicke who had helped to build the instrument with which the 1964 observations were made (Dicke, 1964), and collaborators, also attempted to measure the solar oblateness, using an instrument, SCLERA [Santa Catalina Observatory for Experimental Relativity by Astrometry], which was later to play a role in the early days of helioseismology. This measurement, carried out in 1973, (Hill and Stebbins, 1975), found a 9.6 × 10–6 value for the oblateness, much smaller than that of Dicke and Goldenberg (1967b). Hill et al. (1974) also pointed out a time-varying difference between the brightness of the solar limb and poles that might account for the anomalously high oblateness measurement.

Ulrich and Hawkins (1981a,b) made an early attempt to deduce what the J2 and J4 terms should be based on a simple differential rotation profile deduced from surface measurements, obtaining predicted values of between 1 and 1.5 × 10–7 for J 2 and between 2 and 5 × 10–9 for J 4 depending on the size of the convective envelope.

Dicke et al. (19861987) repeated the 1966 measurements with an improved instrument, and obtained significantly smaller values for the oblateness, with some weak evidence for a solar-cycle variation. Lydon and Sofia (1996) made measurements using a balloon-based instrument and obtained values of 1.8 × 10–7 for J2 and 9.8 × 10–7 for J4. By this point, however, the focus in the solar oblateness studies had moved away from trying to infer the core rotation. Mecheri et al. (2004) used more realistic models of the internal rotation profile to suggest that the J4 term should be particularly sensitive to the subsurface shear. Recent work on determining the oblateness from the shape of the solar limb has taken into account considerations of near-surface temperature or magnetic variations. Kuhn et al. (1998) and Emilio et al. (2007) used observations from MDI during rolls of the SOHO spacecraft and Fivian et al. (2008Jump To The Next Citation Point) used the RHESSI X-ray telescope. The work with SOHO revealed a temporal variation in the shape of the solar limb, with greater apparent oblateness at solar maximum, suggesting that hotter, brighter activity belts have greater apparent diameter. This poses an apparent contradiction to the results obtained from helioseismic inferences of the asphericity. Indeed, Fivian et al. (2008) suggest that all the temporally-varying, excess oblateness found in the observations can be corrected away by removing an ad-hoc term related to magnetic elements in the enhanced network.

Meanwhile, a much more flexible tool – helioseismology – had become available for probing the interior solar rotation.

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