## Consequences

There are many consequences of the magnetic braking during star formation and the MS phase of solar-type stars. Let us consider the case of solid body rotation in the MS phase, where the stellar radius R and moment of inertia I do not change rapidly. The angular momentum is

where v is the rotational velocity at equator. Figure 13.1 illustrates the moment of inertia for stars of various masses and metallicities on the zero-age sequence. The angular momentum of stars of different masses is illustrated as a function of the angular velocity in Fig. 13.2. For stars of standard composition, the moment of inertia behaves with mass like

From (13.33) and (13.38), we get for constant I and R

Since R is taken constant, the integration gives v „ t[-3/(4an) . (13.41)

—■—Z=0.020 —»—Z=0.002 -~* — z= 10"5 • / 0

Fig. 13.1 The moment of inertia for stars of various masses and metallicities Z on the zero-age sequence. From S. Ekstrom et al. 

 yy^ - : / 1MS

Fig. 13.1 The moment of inertia for stars of various masses and metallicities Z on the zero-age sequence. From S. Ekstrom et al.  n/fiont

Fig. 13.2 The values of the angular momentum J of stars of different masses on the ZAMS with Z = 0.02 as a function of a = Q/Qcsit. The changes of the stellar radii with rotation are taken into account. Courtesy from S. Ekstrom

n/fiont

Fig. 13.2 The values of the angular momentum J of stars of different masses on the ZAMS with Z = 0.02 as a function of a = Q/Qcsit. The changes of the stellar radii with rotation are taken into account. Courtesy from S. Ekstrom

For a = 1 and n = 1.5, this gives the Skumanich law  for the decrease of the rotational velocities with time in solar-type stars:

Thus, for n = 1.5 the predicted braking is in agreement with observations for solartype stars. Figure 21.4 shows the evolution of the rotation velocity as a function of age during the pre-MS and MS phases for a 1 M0 star for various n. A curve log v vs. logt intermediate between the curves for n = 1 and n = 2 reproduces well Skumanich's law (13.42) for ages larger than 108 yr. The evolution of the angular velocity is determined by

Thus, both the magnetic braking and the decrease of the moment of inertia I during the MS phase determine the evolution of the rotation velocity of solar-type stars. One has also to account for the transport processes (e.g., meridional circulation), which redistribute the angular momentum.

The change of the parameters of magnetic braking have the following consequences on evolution:

1. Effects of n: larger n values increase the dependence of the loss of J on Q. For larger n, the loss of angular momentum starts earlier during the pre-MS phase. For n = 1.5, the loss of J goes like Q3 (Fig. 21.4).

2. Effects of KW: multiplying the value of KW by a factor f reduces the value of the velocities by f1/2 after the maximum of the curves in the diagram log v vs. logt (Fig. 21.4).

Magnetic braking has many consequences in the formation and evolution of solartype stars (Sect. 21.3).