Unveiling Stellar Cores and Multipole Moments via their Flattening

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.

Appendices

Annex 1: Equation of the Spheroid

The equation of an ellipsoid of revolution is given by:

$$ \frac{r^2(\theta)\,\hbox{cos}^2\theta}{R_{\text{eq}}^2(1-\varepsilon)^2} + \frac{r^2(\theta)\,\hbox{sin}^2\theta}{R_{\text{eq}}^2} = 1 $$

(5.40)

Expanding $r(\theta)$ in powers of $\varepsilon,$ we obtain

$$ r(\theta) = R_{\text{eq}}\left[1 -\varepsilon \hbox{cos}^2 \theta - \frac{3}{2} \varepsilon^2(\hbox{sin}^2\theta \,\hbox{cos}^2\theta) \right. + \left. \frac{1}{8}\varepsilon^3(1-5 \hbox{sin}^2\theta) \, \hbox{sin}^2 2\theta + \cdots \right] $$

(5.41)

Designing by $k(R_{\text{eq}})$ and $h(R_{\text{eq}})$ the second-order and third-order corrections, (5.41) becomes ⁸

$$ \begin{aligned} r(\theta) &= R_{\text{eq}}\left[1 -\varepsilon \hbox{cos}^2 \theta - (\frac{3}{8} \varepsilon^2 + k) \hbox{sin}^2 2\theta \right. \\ &\left.\,\,\quad\qquad\qquad + \frac{1}{4} ( \frac{1}{2}\varepsilon^3 + h )(1 - 5 \hbox{sin}^2\theta) \,\hbox{sin}^2 2\theta + \cdots \right]. \end{aligned} $$

(5.42)

Inserting this development of the radius vector $r(\theta)$ in the expression of the total potential one obtains

$$ \Upphi_{\text{tot}} = -\frac{GM}{R_{\text{eq}}}\left[ 1 + a(\varepsilon,\theta) + b(\varepsilon,\theta) J_2 + c(\varepsilon,\theta) J_4 + \cdots \right] $$

(5.43)

$J_n$ 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

$$ \Upphi (\theta) = \frac{\Upomega^2 R^2(\theta) \hbox{sin}^2 (\theta)}{2} + \frac{GM}{R(\theta) }= \frac{GM}{R_{\text{pol}}} $$

(5.44)

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

$$ r^3(\theta) -r(\theta) \left(\frac{1}{1-D} \right) \frac{1}{\hbox{sin}^2(\theta)} + \left( \frac{D}{1-D} \right) \frac{1}{\hbox{sin}^2(\theta)} = 0 $$

(5.45)

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:

$$ \frac{R_{\rm c}}{R_{\rm p}} = \frac{3}{2} \equiv D_{\rm c} $$

$$ v_{\rm c}=\Upomega_{\rm c} R_{\rm c}=\sqrt{\frac{GM}{R_{\rm c}}}=\sqrt{GM\left( \frac{2}{3R_{\rm p} }\right)} $$

$$ \frac{v_{\rm eq}}{v_{\rm c}}=\sqrt {3\left( 1-D\right)} = \frac{\Upomega}{\Upomega_{\rm c}} \frac{R_{\rm eq}}{R_{\rm c}} \equiv \omega \frac{R_{\rm eq}}{R_{\rm c}} $$

$$ \Upomega_{{\rm c}}= \frac{v_{\rm c}}{R_{\rm c}}=\sqrt{\frac{8GM}{27{R_{\rm p}^{3}}}} $$

$$ \frac{\Upomega }{\Upomega _{\rm c}}= \omega =\sqrt{3( 1-D)}\left( \frac{3}{2}D\right) $$

One will note that the angular ($\Upomega)$ and linear ($v$) velocities are linked by $\Upomega = v/R (\hbox{sin}\,i).$