Relation between stellar rotation and mass?

[publius]

Is there any simple relation (approximate or otherwise) between the angular momentum/rotation rate of stars vs their mass and radius?

I ask because of this thread:

http://www.bautforum.com/showthread.php?t=45728

Grav "stumbled" upon a formula for orbital precession of Mercury and other planets that agrees closely with GR. He postulated that the orbital precession (in fractions of a revolution per revolution, "orbits per orbit" roughly) was given by v/c*R/d, where R is the radius of the sun, and 'd' is the rough radius of an orbit. Now, guess what? For the sun that comes out to be very close to GR's (approximate, dropping higher order terms that are negligible for solar system orbits) formula for the same as

3*r_s/L, where r_s is the Schwarzchild radius, and L is the "semilatus rectum", which for near circular orbits will be about 'd' as well.

Taking the ratio of grav's coefficient vs 3*r_s, one gets this:

c/6G * vR/M

Now, vR is roughly proportional to the specific angular momentum (to "zeroth order", not paying attention to variations in density, differential rotation and all that good stuff), so Grav's ratio is propotional to ratio of specific angular momentum to mass, or angular momentum per mass squared.

For the sun, that times c/6g is ~ 1. I would like to know why this ratio is so close to 1 for the sun. Pure coincidence or is there some physics at work here?

-Richard

[Ken G]

It's pure coincidence, the rotation speed does not enter into the gravity, but the Schwarzschild radius does. Grav's ratio is also of order c*v/u^2, where u is the escape speed from the surface and v is the rotation speed (I presume), or v/c times the ratio of the Sun's rest energy to its binding energy. The second is a measure of how nonrelativistic of an object the Sun is (very nonrelativistic), and the former has nothing to do with relativity-- relativity never appears to first order in v/c. So the two have nothing to do with each other, and indeed v drops rapidly with age for solarlike stars, though none of the other parameters do.

[publius]

Ken,

Thanks. What I was actually wondering if some model of stellar formation predicted a relationship between mass and angular momentum, and if that might shed some light on why grav's ratio is so close for the sun. Sort of an explanation for the coincidence.



-Richard

[Ken G]

Yes, I realize this was your goal, but the fact that solarlike stars quickly slow their rotation as they age would preclude such a general relation. Low mass stars typically are slow rotators, this is true, because they are believed to be braked by magnetic fields as they form. High mass stars are able to be rapid rotators, and are more likely to be close to their critical rotation rate than low mass stars. If you consider critical rotation, then v ~ u, so vR/M scales like 1/u. So if all stars were critical rotators, then main sequence stars (which all have similar u) would have similar values of the ratio you mention. But many stars (like the Sun) are nowhere close to being critical rotators, while others are, so the relation doesn't even hold along the main sequence. It's a coincidence for the Sun, that's all.

[publius]

Ken,

I see. Rotation rate is such a "wild" variable that it is indeed pure coincidence. I guess its sort of like the moon being about the same angular size as the sun in the sky, allowing for spectacular full eclipses.

-Richard

[Ken G]

Yeah, I think so, coincidences are everywhere, and don't all need to be explained.

[hhEb09'1]

I see. Rotation rate is such a "wild" variable that it is indeed pure coincidence. I guess its sort of like the moon being about the same angular size as the sun in the sky, allowing for spectacular full eclipses.

There's also that the Sun's rotation is not a pure rotation, either. The speeds of the surface regions vary, don't they?

[Peter Wilson]

Yes, the equator rotates slightly faster than the poles, for reasons unclear to me.

Also, denser materials tend to "sink" to the center, carrying angular momentum with them as the do so, thus the core rotates faster than the outer envelope.

[Ken G]

Also, denser materials tend to "sink" to the center, carrying angular momentum with them as the do so, thus the core rotates faster than the outer envelope.

No, that just isn't true.

[Romanus]

Probably, but not for the reasons mentioned.

Early-type stars (e.g., massive) stars often rotate much more rapidly than the Sun, but this is an artifact of their different structures; IIRC, as a general rule they have weaker magnetic fields due to a lack of an external convection zone, which decreases the magnetic "braking" a star's rotation.

[George]

It has been my understanding the core and radiative zones rotate as a rigid body, whereas the convective zone has differential rotation (25 days at the equator vs. 35 days at the poles for the photosphere). The zone where fixed rotation and differential rotation meet is called the tachocline. It is suspected sunspots take root here.

[Ken G]

Probably, but not for the reasons mentioned.

See post #4.