Stellar Surface and Gravity

The stellar surface is an equipotential W = const., otherwise there would be "mountains" on the star and matter flowing from higher to lower levels. The total potential at a level r and at colatitude $ ($ = 0 at the pole) in a star of constant angular velocity Q can be written as

GMr 1 T T T W(r, $) =--j- — ^ Q2 r2 sin2 $. (2.9)

One assumes in the Roche model that the gravitational potential 0 = —GMr/r of the mass Mr inside radius r is not distorted by rotation. The inner layers are considered as spherical, which gives the same external potential as if the whole mass is concentrated at the center.

Let us consider a star of total mass M and call R($) the stellar radius at colatitude $. Since the centrifugal force is zero at the pole, the potential at the stellar pole is just GM/Rp, where Rp is the polar radius. This fixes the constant value of the equipotential at the stellar surface, which is given by

A more tractable form is given below (2.18). The shape of a Roche model is illustrated in Fig. 2.2 for different rotation velocities (the radii for non-rotating stars of different masses and metallicities Z are given in Fig. 25.7). Figure 2.3 illustrates the variation of the ratio of the equatorial radius to the polar radius for the Roche model as a function of the parameter co = Q/Qcrit. We see that up to co = 0.7, the increase of the equatorial radius is inferior to 10%. The increase of the equatorial radius essentially occurs in the high rotation domain.

The effective gravity resulting from the gravitational potential and from the centrifugal force is given by (2.8). If er and e$ are the unity vectors in the radial and latitudinal directions, the effective gravity vector at the stellar surface is

Fig. 2.2 The shape R(tf) of a rotating star in one quadrant. A 20 M® star with Z = 0.02 on the ZAMS is considered with various ratios a = Q/Qan of the angular velocity to the critical value at the surface. One barely notices the small decrease of the polar radii for higher rotation velocities (cf. Fig. 2.7). Courtesy of S. Ekstrom

Fig. 2.2 The shape R(tf) of a rotating star in one quadrant. A 20 M® star with Z = 0.02 on the ZAMS is considered with various ratios a = Q/Qan of the angular velocity to the critical value at the surface. One barely notices the small decrease of the polar radii for higher rotation velocities (cf. Fig. 2.7). Courtesy of S. Ekstrom

Fig. 2.3 The variation of the ratio Re/Rp of the equatorial to the polar radius as a function of the rotation parameter a in the Roche model

Fig. 2.3 The variation of the ratio Re/Rp of the equatorial to the polar radius as a function of the rotation parameter a in the Roche model er + Q2R(tf) sintf cos tf] . (2.11)

geff

The gravity vector is not parallel to the vector radius as shown in Fig. 2.1. The modulus geff = | geff | of the effective gravity is

geff

which can also be written as in (2.20). 2.1.4 Critical Velocities

The critical velocity, also called break-up velocity, is reached when the modulus of the centrifugal force becomes equal to the modulus of the gravitational attraction at the equator. The maximum angular velocity Qcrit, which makes geff = 0 at the equator (tf = n/2) is thus from (2.12)

Re,crit where Re,crit is the equatorial radius at break-up. If one introduces this value of Qcrit in the equation of the surface (2.10) at break-up, one gets for the ratio of the equatorial to the polar radius at critical velocity,

Rp,crit 2

At break-up, the equatorial radius is equal to 1.5 times the polar radius. The equatorial break-up velocity is thus

Re,crit 3 Rp

This expression is the one quite generally used; however, formally it applies to solid body rotation. The index "1" indicates the classical critical velocity, to distinguish it from a second value ucrit,2 which applies to high mass stars with a high Eddington factor (see Sect. 4.4.2). If we now introduce a non-dimensional rotation parameter co, defined as the ratio of the angular velocity to the angular velocity at break-up,

Q 2 Q 2 Re3,crit co = —— which gives co2 = ———-— . (2.16)

Qcrit GM

One can also write c 2

p crit and the equation of the surface (2.10) becomes with x = R/Rp,crit

If the polar radius does not change with co (but see Fig. 2.7), the second member is equal to 1. Equation (2.18) is an algebraic equation of the third degree. Our experience suggests it is better solved by the Newton method rather than by the Cardan solutions of a third degree polynomial, because the Cardan solutions may diverge at the poles. Also depending on the value of Rp.crit/Rp(®), for co close to 1 the discriminant of the polynomial equation may change its sign implying another form for the solution. The comparison with the interferometric observations of stellar oblateness and gravity darkening is given in Sect. 4.2.3.

Figure 2.4 shows the critical velocities vcrit.i for stars of various masses and metallicities. The critical velocities grow with stellar masses, because the stellar radii increase only slowly with stellar masses. The critical velocities are very large for low metallicity stars, since their radii are much smaller as a result of their lower opacities. Comparison between observed and theoretical values of velocities is given in Table 4.1. The distribution of rotational velocities for about 500 B-type stars is shown in Fig. 27.1.

Figure 2.5 shows the ratio v/vcrit.i of the equatorial velocity to the critical equatorial velocity as a function of the parameter c — Q /.Ocrit in the Roche model. For low rotation (c < 0.5), a linear approximation of v in terms of co is valid

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