Pulsar size vs. frequency

[theridane]

Imagine a massive spinning star nearing the end of its life. As it goes supernova, a pulsar is formed. Conservation of momentum makes this pulsar spin up to tremendous angular rates (e.g. the PSR J1748-2446ad spinning at 716 Hz).

Speed of light plays an intersting role in this. The equator of a pulsar reaches relativistic velocities, which means that there is a limit to pulsar's size and/or rotational rate (its radius must be less than c/(2*pi*frequency), otherwise it would be travelling FTL, which is a bad thing for an equator to do).

My question is the following: if the mother star was large/spinning fast enough so that the resulting pulsar would violate the above equation, how would "the universe" respond? Would the pulsar slow down (as it forms) or would it undergo some sort of a "relativistic compression" event that would reduce its radius to satisfy the equation?

I'm guessing both, since the relativistic mass gain would be so huge that it would shrink the star and increase its momentum, allowing it to conserve it even at a lower rotational frequency.

 

[emperor_of_localgroup]

Imagine a massive spinning star nearing the end of its life. As it goes supernova, a pulsar is formed. Conservation of momentum makes this pulsar spin up to tremendous angular rates (e.g. the PSR J1748-2446ad spinning at 716 Hz).

Speed of light plays an intersting role in this. The equator of a pulsar reaches relativistic velocities, which means that there is a limit to pulsar's size and/or rotational rate (its radius must be less than c/(2*pi*frequency), otherwise it would be travelling FTL, which is a bad thing for an equator to do).

I crunched some non-relativistic numbers for the size you gave in your post. With that size (radius), to keep a small chunk of matter on its surface, the mass of the pulsar must be
M = 9x10[super]31[/super] Kg. I used frequency f = 716Hz for radius
R = 66685 m
The density of pulsar matter with this numbers turns out to be
density = 7.24x10[super]16[/super]Kg/m[super]3[/super]
Density of a proton is 60x10[super]16[/super]Kg/m[super]3[/super]. Only 8.3 times higher than pulsar matter. So the question is are pulsars a new type of matter? Only deformed protons can create such density. For this and the assumption that spacetime and inertia have no effect on high speed rotation of massive objects I have always accepted pulsar rotation theory with some doubt.

 

[theridane]

I've gone over your figures and they check out, except for proton density. I was unable to find any major reference to that term - I'm guessing you used a proton's charge radius to calculate its volume (wrongly assuming that protons are teeny tiny spheres) and used that to divide its rest mass, right?

Neutron stars have densities in the 37e16 to 59e16 kg/m³ ballpark, so the pulsar in your calculation is in fact less dense (7.24e16) than a real one. Turns out there are other constraints on its size than just special relativity - the 716Hz example is thought to be just 16 km in radius.

 

[csmyth3025]

I posted the following in another physics thread but, because of the similar subject matter, I think this thread is probably more appropriate:

Is there a theoretical limit to the angular momentum (rotational speed) of a black hole?

For the purpose of this question lets use a black hole of about the size of the one that's believed to reside at the center of the Milky Way (about 4 million solar masses).

I know there is a Kerr metric which is an exact solution for the Einstein field equations of General Relativity as applied to uncharged rotating bodies. I've read about this metric but I don't pretend to understand it even a little bit.

My question stems from the Wikipedia article on the Kerr metric which can be found here:

http://en.wikipedia.org/wiki/Kerr_metri ... se_process

The specific portion which prompts my question is as follows:

A rotating black hole has the same static limit at the Schwarzschild radius but there is an additional surface outside the Schwarzschild radius named the "ergosurface" given by (r − GM)2 = G2M2 − J2cos2θ in Boyer-Lindquist coordinates, which can be intuitively characterized as the sphere where "the rotational velocity of the surrounding space" is dragged along with the velocity of light. Within this sphere the dragging is greater than the speed of light, and any observer/particle is forced to co-rotate.

To me, as a layman, this portion seems to be saying that there is a theoretical exotic object that is capable of "dragging" a particle faster than the speed of light in relation to a distant observer in the "outside" universe. As I understand it the particle isn't moving faster than the speed of light relative to the spacetime of the ergosphere in which it's imbedded.

Using the WolframAlpha website, which can be found here: http://www.wolframalpha.com/
I calculate that the central black hole of 4 million solar masses will have a circumference at the Schwarzschild radius of 7.42 x 10^7 km (for a non-rotating black hole). If this object makes more than one revolution every ~4 minutes, the spacetime of the ergosurface will be rotating at ~300,000 km/sec (at the "equator").

I suppose my question can be broken down to two parts: First, are there theoretical considerations which would prevent such a rotating black hole from forming? Second, has anyone been able to measure or estimate the rotational speed of an actual black hole?

Chris

 

[emperor_of_localgroup]

I've gone over your figures and they check out, except for proton density. I was unable to find any major reference to that term - I'm guessing you used a proton's charge radius to calculate its volume (wrongly assuming that protons are teeny tiny spheres) and used that to divide its rest mass, right?

Neutron stars have densities in the 37e16 to 59e16 kg/m³ ballpark, so the pulsar in your calculation is in fact less dense (7.24e16) than a real one. Turns out there are other constraints on its size than just special relativity - the 716Hz example is thought to be just 16 km in radius.

In fact I calculated all the numbers in my post except one, the density of proton, which I found in a website, which I think was calculated by using the procedure you described. If neutron stars have density of 37e16 to 59e16 kg/m³, my number 60e16 kg/m³ for proton density, which is lower than neutron stars, seems to be right.

The bottom line of my doubt is pulsar's flashing frequency may not be as linearly related to its rotational frequency as current theories predict. Several flashing may be occurring for one spin.

CSMYTH has pointed out another dilemma. If pulsars are rotating with such high angular velocity, then why not neutron stars and black holes? Their rotational speeds may even be much higher.

which can be intuitively characterized as the sphere where "the rotational velocity of the surrounding space" is dragged along with the velocity of light. Within this sphere the dragging is greater than the speed of light, and any observer/particle is forced to co-rotate.

A massive object with stronger gravity may rotate (drag) the space-time itself.

I calculate that the central black hole of 4 million solar masses will have a circumference at the Schwarzschild radius of 7.42 x 10^7 km (for a non-rotating black hole). If this object makes more than one revolution every ~4 minutes, the spacetime of the ergosurface will be rotating at ~300,000 km/sec (at the "equator").

Here we are talking about 716 revolutions in 1 SEC. I am assuming all stars have spins, then it is safe to say all black holes are also spinning and with much higher speed than when they were stars. If the space-time around a black hole is being dragged with such speed, then how can anything, including light, fall into a black hole?

What am I missing in my thought process?

 

[csmyth3025]

I found the following information on neutron stars in the Wikipedia article on this subject here:

http://en.wikipedia.org/wiki/Neutron_star

The pertinent portion to this discussion is in the second paragraph, as follows"

A typical neutron star has a mass between 1.35 and about 2.1 solar masses, with a corresponding radius of about 12 km if the Akmal-Pandharipande-Ravenhall (APR) Equation of state (EOS) is used.[1][2] In contrast, the Sun's radius is about 60,000 times that. Neutron stars have overall densities predicted by the APR EOS of 3.7×10^17 to 5.9×10^17 kg/m3 (2.6×10^14 to 4.1×10^14 times the density of the Sun),[3] which compares with the approximate density of an atomic nucleus of 3×10^17 kg/m3.[4] The neutron star's density varies from below 1×10^9 kg/m3 in the crust increasing with depth to above 6×10^17 or 8×10^17 kg/m3 deeper inside.[5]

A description of the probable rotational period of neutron stars is also contained further down in this same article here:

http://en.wikipedia.org/wiki/Neutron_star#Rotation

Neutron stars rotate extremely rapidly after their creation due to the conservation of angular momentum; like spinning ice skaters pulling in their arms, the slow rotation of the original star's core speeds up as it shrinks. A newborn neutron star can rotate several times a second; sometimes, the neutron star absorbs orbiting matter from a companion star, increasing the rotation to several hundred times per second, reshaping the neutron star into an oblate spheroid.

My related question about the rotation of a black hole and the Kerr metric is prompted by the thought that if a star with a core remnant of 1.3 to 2.1 solar masses can collapse to a neutron star with a radius of ~12 km and a rotation of "several times a second", then a star with a core remnant of ~5 solar masses will "inevitably collapse into a black hole" with a radius of (?) [lets say ~1.2 mm for the "singularity" - which would be 10^-7 that of a neutron star]. This would result in the singularity having a rotation of several million times a second for a normally rotating progenitor star.

If this black hole can exist as postulated, then I think the event horizon would co-rotate with it - somewhat like the so-called "theoretical rigid disk". This would be a strange object, indeed. It would, I think, be the "naked singularity" that some have proposed.

I've already asked if there are theoretical considerations which would prevent such an object from forming. I'm also wondering if the rotation of a black hole has ever been estimated or measured. In regard to the formation of "naked singularities", there's a Scientific American article (Feb., 2009 issue) which suggests that "naked singularities" may not be so very unusual. It can be found here:

http://www.scientificamerican.com/artic ... gularities

Another question I have is: If this object exhibits "frame dragging", would all the matter falling into it acquire an angular velocity co-rotating with the black hole and, thus, increase the angular momentum (rotation) of the singularity?

Chris