Abstract
Rotation, and more precisely differential rotation, has a major impact on the internal dynamics of stars (and the Sun) and induces many instabilities driving the transport of angular momentum. In this chapter we shall consider these effects on the shape of shelllular layers, and to first order, those concerning the apparent oblateness. Thanks to the advent of interferometry techniques, stellar shapes can now be measured with a great accuracy. We will review here some main results obtained so far on different stars and we will give their main physical parameters taking into account differential rotation. We will discuss how the core density can be reached. Gravitational moments are presented for these observed flattened stars, and for the Sun, for which some conflicting results are presented.
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Notes
- 1.
The Von Zeipel’s [24] theorem stipulates that contours of temperature, density, or pressure should be nearly coincident near the surface. Differential rotation, magnetic fields and turbulent pressure are the largest local acceleration sources that may violate this theorem. It has been generalized to account for differential rotation in the case of a “shellular” rotation law by Maeder [36].
- 2.
Note the relation \(-s_0 =\frac{1}{5} s_2 + \frac{2}{105} s_2^3.\)
- 3.
Helioseismic inversions provide a more realistic rotation profile, albeit less practical, that we shall recall here. Assuming a solid rotation below 0.66, and a differential rotation above the interface, one can write:
$$ \Upomega(r, \theta) = \Upomega_c + \frac{1}{2}\big[1 + \mbox{\rm erf}(2*\frac{r-r_c}{d_1}\big] \times (\Upomega_{Eq} + a_2\hbox{\hbox{cos}}^2 \theta + a_4 \hbox{\hbox{cos}}^4 \theta - \Upomega_c), $$(5.22)with \(\Upomega_{Eq}=1,\) \(\Upomega_c=0.93944,\) \(r_c=0.7,\) \(d_1=0.05,\) \(a_2=-0.136076\) and \(a_4=-0.145713;\) “erf” is the error function. With this profile, the radial shear is maximal at the tachocline. Using this profile and numerically integrating the equipotentials from the core to the surface leads to a \(J_2\) lying between 1.60 \(10^{-7}\) and 2.20 \(10^{-7}\) [34].
- 4.
There is a sign mistake in [7] p. 23; the slight difference in the estimates is due to the different values used for the solar radius.
- 5.$$ \hbox{given by}\quad \frac{d^2f}{dq^2}+ \frac{6}{q}\frac{\rho}{D}\frac{f}{q} - {\frac{6}{q^{2}}}\left(1-\frac{\rho}{D}\right)f = 0, \quad \hbox{where} \;{\it D}\;\hbox{is}\; \frac{3}{q^3}\int\limits_0^q \rho q^2 dq. $$
- 6.
Or 1.9891 \(\times 10^{33}\) \({\rm g}\) \(\pm\) 0.0004 (Cohen and Taylor, 1984).
- 7.
The estimate based on surface rotation alone [46] is 1.63\(\times10^{+48},\) which is about 15% smaller. The average angular momentum as a function of radius (and latitude) basically determined by the solar model that provides the density as a function of radius.
- 8.
Truncated at the order 3, and putting \(h= 0,\) (5.42) represents the so-called Darwin-de Sitter spheroid equation.
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Appendices
Annex 1: Equation of the Spheroid
The equation of an ellipsoid of revolution is given by:
Expanding \(r(\theta)\) in powers of \(\varepsilon,\) we obtain
Designing by \(k(R_{\text{eq}})\) and \(h(R_{\text{eq}})\) the second-order and third-order corrections, (5.41) becomes Footnote 8
Inserting this development of the radius vector \(r(\theta)\) in the expression of the total potential one obtains
\(J_n\) represents the gravitational moments(see 5.7). On the equilibrium surface, \(\Upphi_{\text{tot}}\) must be constant (i.e. independent from \(\theta),\) so that the \(J_n\) coefficients must vanish (see 5.14).
Annex 2: Roche’s Model
The stellar equipotential surfaces of a body of mass \(M\) rotating at constant angular velocity \(\Upomega\) are described by
Introducing the degree of sphericity \(D = R_{\text{pol}}/R_{\text{eq}},\) which relates the inverse of the oblateness to the polar radius, (5.44) can be rewritten as
where \(r(\theta)\) designs the normalized radius \({\equiv} R(\theta)/R_{\text{eq}}.\) Solution of (5.45) is obtained through hypergeometric series of argument \(\varsigma\) given by \( \varsigma^2 \equiv 2 \frac{(1-D)}{D} (\frac{3D}{2})^3 \hbox{sin}^2 \theta\) [47], which determine the stellar shape. Lastly, it can be useful to remember the quantities relating critical and non-critical parameters. The critical or break-up velocity for the Roche’s model is attained when the centrifugal and gravitational forces are equal. this leads to the following set of equations:
One will note that the angular (\(\Upomega)\) and linear (\(v\)) velocities are linked by \(\Upomega = v/R (\hbox{sin}\,i).\)
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Rozelot, JP., Damiani, C., Kilcik, A., Tayoglu, B., Lefebvre, S. (2011). Unveiling Stellar Cores and Multipole Moments via their Flattening. In: Rozelot, JP., Neiner, C. (eds) The Pulsations of the Sun and the Stars. Lecture Notes in Physics, vol 832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19928-8_5
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