Tuesday, November 13, 2012
Solar Oblateness May Be Constant Over Time
As with all rotating celestial bodies - planets included - the Sun is "oblate", meaning that it "bulges" at the waist, i.e. solar equator. The measure of this bulge is then the ratio of the equatorial diameter's excess (i.e. over the polar dia. or (a- b)) to the polar diameter (b) itself. Or:
Obl = (a - b)/ a
Solar oblateness came to the fore some decades ago when it was invoked to challenge the standard Einsteinian General Relativity. This was as part of the Jordan- Brans- Dicke theory of gravity proposed by Robert Dicke of Princeton and Carl Brans - my former mathematical physics prof at Loyola University ca. 1966.(See, e.g. http://chn.loyno.edu/physics/bio/carl-h-brans)
What Brans and Dicke hoped to find, to validate their theory, was a departure from spherical symmetry large enough that it had the potential to affect Mercury's orbit. This quantity, dubbed 'eta' by Einstein, referred to the advance of the perihelion in seconds of arc, over a century. It is instructive to first revisit this and show how Einstein obtained his value of 43 seconds of arc per century.
Thus: eta = [24 (π)^3* a^2]/ T^2 *c^2*(1 - e^2)
where 'eta' is the advance of perihelion in seconds of arc, T is the period of revolution in seconds, c the velocity of light in cm/s and e the eccentricity of the orbit. From a Table of distances, Mercury's semi-major axis = 0.387 AU or: a = (1.5 x 10^13 cm) (0.387) = 5.8 x 10^12 cm
The period, T (in seconds) is just the length of Mercury's year (in days = 87.96, again from a Table) multiplied by the seconds-length of an Earth day, or 86,400 s:
T = 7.6 x 10^6 s
The eccentricity, e from a similar Table is e = 0.205.
Substituting all these values into the given equation yields: eta = 5.036 x 10^-7 radian
However, this is not in seconds of arc but radian measure. To get the equivalent seconds of arc (or arcsec) we use the fact that 1 rad (radian) = 57.3 deg and one degree has 3600 seconds. Thus, 1 radian = 2.063 x 10^5 arcsec
So, the associated arcsec for eta will be 5.036 x 10^-7 rad) x (2.063 x 10^5 arcsec/ rad) =0.104 arcsec
We are still not finished because the quantity is defined per CENTURY
At this point, you need to recall the PERIOD of Mercury is 0.24085 YRS.
So, the number of arcsec of perihelion advance per Earth years is:
0.104 arcsec/ 0.24085 years = 0.431
and over 100 years:
eta = 100 yr x (0.431 arcsec/ yr) ~ 43.1 arcsec, compared to the value of 42.98 "/century predicted by Einstein.
Interrelated with the above is the fact that an oblateness gives the Sun what is called a "quadrupole moment" which we designate J2. The dimensionless parameter J2 is defined (cf. 'Gravitation and Spacetime' by Hans C., Ohanian and Remo Ruffini, W.W. Norton, p.18):
J2 = Q^33/ 2M R^2
where M is the mass of the Sun, R is its radius, and Q^kl = Q^33 is the mass quadrupole tensor (op. cit., p. 17). To assess a baseline perspective, if the Sun were rotating totally uniformly oblateness effects minimal) then J2 would be roughly 1 x 10^-7. In this case, the corresponding contribution to the advance of Mercury's perihelion would be a few hundredths of an arcscond per century, or in other words, within the observed value of the uncertainty for 'eta'.
Interestingly, Brans and Dicke (1961), originally proposed a 'scalar-tensor' theory of gravitation which avoided conflict with the Einstein GR prediction. In their field the scalar field replaced the gravitational constant and becomes instead a function of space and time, so oe gets a variable value for G - the Newtonian gravitational constant. Effectively, on is looking at a quantity G'/G where G' denotes the first time derivative, or dG/dt/
Meanwhile, in 1967, Dicke and Goldenburg obtained an unexpectedly large J2 ~ 2.4 x 10^-5 which would show significant deviation from the Einstein eta. It would also have required that the solar interior rotate at a much faster rate than the surface.Such a large quadripole moment would have led to an additional 3 arcsecs per century over Einstein's value. Thus, leading to eta ~ 46"/ century. This large value has never been confirmed, and indeed, Dicke et al (1985) using a solar vibration method found J2 = 1.7 x 10^-7.
Quite obviously, what the preceding showed is that better methods of measuring oblateness (and hence the quadripole moment J2) were needed, which exceeded optical methods, including digital photocells, and also oscillatory -vibration methods. Such an advance on observational quality became feasible in 2010 with the launch of NASA's Solar Dynamics Observatory (SDO). This Observatory was able to measure with record precision the shape of the solar limb (the solar disk's edge) and in addition ascertain variations in the limb over a solar cycle.
The results forthcoming were simply amazing, i.e. that the Sun's equatorial radius exceeded its poalr radius by 7.2 + 0.5 milliarcseconds, and with NO evidence of departure from constancy. Given the Sun's diameter of 1.4 miilion kilometers subtends a half degree in the sky (1800" roughly) then the difference between its equatorial and polar radii would amount to barely 5 km. In other words, it's almost perfectly spherical. Most remarkably, the 5 km difference is small even when compared to the turbulence on the solar surface
Further refinement - especially useful over solar cycles - was obntained by incorporating best -fit quadrupole (C2) and hexadecapole (C4) coefficients. The to 4th order the difference between equatorial and polar radii is given by:
D = -3C2/ 2 - 5C4/8
Amazingly, within the range of observational uncertainties, both C2 and C4 have been consistent with constancy over two years. Moreover, C4 is effectively zero (contributes very little to solar oblateness) so:
D = -3C2/ 2
Further analysis using Legendre polynomial multipole data discloses a marginal hint of correlation between C4 and the sunspot cycle - which may or may not assume importance for future sunspot -flare forecasts. This association, for the time being, is theorized to be "a result of magnetic stresses localized at mid-latitudes or in near surface flows". (Physics Today, Oct., p. 14).
Of course, we must not be too hasty in drawing firm conclusions about constancy of oblateness from just two years of observation. But thus far, it looks pretty good.