tusenfem
Illuminator
- Joined
- May 27, 2008
- Messages
- 3,306
Okay, I have no problems with the graph. That is what it is, and I have seen it in classes at university. However, the ideas of Alexeff are a bit weird, IMHO.
My bold. The nature of the operator is not known? Does Alexeff know his Maxwell equations? That differential operator is just a combination of spatial derivatives, and cannot be anything else. Here is the definition of curl .
Okay, stop it here for a moment. Equation (3) comes, when I read it correctly from J = σ E, with E the electric field. Now, it is stated that the current must be created by the rotation of the planet. Well, one could do this, if the planet is charged and rotating, you will have a current. However, that is not what Alexeff is doing, no, he takes the motional electric field and states that E = v × B. Nice, but not appropriate, because there is no B. This is where a referee should have jumped in (but being published in IEEE the referees don't have up much with plasma physical equations).
So, in order to get a current one needs an ambient magnetic field, so B in equation 2 and B in equation 3 are not the same magnetic fields!
Yeah, I would say this is ambiguous, because if you take out B on both sides, you find a nice expression for the radius of a star/planet/galaxy:
R2 = ( μ0σω )-1
wowie!
Once more, what magnetic field??? Alexeff is assuming an a priory magnetic field in which the planet is embedded!
Now, I don't know what Bohm conduction is, but looking at the equations Alexeff means that a current is made of of moving electrons, sure, why not. However, now it becomes interesting, because there are also two v's.
The current is defined as σE (fine) and as eneν (also fine). Then in the next =-sign the velocity of the electrons is set to the velocity in E = v × B velocity. However, do we know that the planet is charged? Where is the initial assumption that there is charge on the planet? I did not see any, maybe Zeuzzz forgot to copy that part of the model?
And then the last step is just writing the E = v × B velocity (deduced from E/B) to ωR. All good and fine, but only works if the rotation axis of the charged planet is parallel to the background field.
Naturally, one could just look at the magnetic field of a spinning charged sphere, which would be easier that all this ambiguous stuff here, and probably easily looked up in Jackson (which I don't have at home, where I am now).
So, now he puts in the J from equation 5 into equation 2. Still this could maybe work if the planet is charged and if there is already a background magnetic field in the plasma in which the planet is embedded and I am still sure that there is a confusion of two B's which are confusingly not labeled.
This kind of definition for magnetic moment I have never seen.
Normally, the magnetic field of a magnetic moment m is given as:
B = (mu0 / 4 pi) (3 n(m.n - m) / r3
where n is a unit vector in direction r and m is the magnetic moment of a dipole (of which Alexeff is talking I think). Naturally, as Alexeff is not to thrilled about differentiation, integration and vectors, he can easily invert the above equation for the poles of the planet. Maybe the same comes out. Actually, mu0 seems to fall out then, typo by Zeuzzz or an error by Alexeff?
Sure why not, interesting detail, we are dealing with a cylinder??? Why? Is the angular momentum for a sphere too complicated to calculate? Anywho: L = MωR2 for a particle at R of mass M and angular velocity ω. For a sphere of constant density (why not, life is easy) we can quickly integrate L for the total angular momentum
Ltot = int int int MωR2 2R2 dR dphi dtheta,
where M and ω are constants (actually M is then a density but whatever) and doing the integration one finds that
Ltot = M ω (4 pi2/3) R5
Ah well, the same sortof.
Well, indeed in this calculation R falls out of the ratio. However, this "calculation" is so riddled in unclarities from the start, that it is hard to know whether this has any significance on the graph or not.
"We now develop a very simplified model of the mechanism of magnetic field generation, glossing over details in integration in order to present the basic details of the model. Consider Maxwell’s equations. The magnetic field is generated by a current
∇×B = μ0J. (1)
The magnetic field is operated on by a differential operator. The exact nature of the operator is not known at present, but the basic spatial distance in the differentiation process is obviously the celestial body’s radius R. Hence
B/R = μ0J. (2)
My bold. The nature of the operator is not known? Does Alexeff know his Maxwell equations? That differential operator is just a combination of spatial derivatives, and cannot be anything else. Here is the definition of curl .
Next, the current must be generated by the rotation of the planet
J = σVB = σωRB. (3)
Okay, stop it here for a moment. Equation (3) comes, when I read it correctly from J = σ E, with E the electric field. Now, it is stated that the current must be created by the rotation of the planet. Well, one could do this, if the planet is charged and rotating, you will have a current. However, that is not what Alexeff is doing, no, he takes the motional electric field and states that E = v × B. Nice, but not appropriate, because there is no B. This is where a referee should have jumped in (but being published in IEEE the referees don't have up much with plasma physical equations).
So, in order to get a current one needs an ambient magnetic field, so B in equation 2 and B in equation 3 are not the same magnetic fields!
Equating the two expressions produces a result that is ambiguous in B
B/R = μ0σωRB (4)
Yeah, I would say this is ambiguous, because if you take out B on both sides, you find a nice expression for the radius of a star/planet/galaxy:
R2 = ( μ0σω )-1
wowie!
However I note that the current flow from the equator of the celestial body is across the magnetic field in plasma, and Bohm conduction is appropriate. Bohm conduction will be discussed in more detail later [....]
J = σE = eneν = ene(E/B) = eneωR. (5)
Once more, what magnetic field??? Alexeff is assuming an a priory magnetic field in which the planet is embedded!
Now, I don't know what Bohm conduction is, but looking at the equations Alexeff means that a current is made of of moving electrons, sure, why not. However, now it becomes interesting, because there are also two v's.
The current is defined as σE (fine) and as eneν (also fine). Then in the next =-sign the velocity of the electrons is set to the velocity in E = v × B velocity. However, do we know that the planet is charged? Where is the initial assumption that there is charge on the planet? I did not see any, maybe Zeuzzz forgot to copy that part of the model?
And then the last step is just writing the E = v × B velocity (deduced from E/B) to ωR. All good and fine, but only works if the rotation axis of the charged planet is parallel to the background field.
Naturally, one could just look at the magnetic field of a spinning charged sphere, which would be easier that all this ambiguous stuff here, and probably easily looked up in Jackson (which I don't have at home, where I am now).
The net result is as follows:
B = μ0eneωR2. (6)
Thus, the magnetic field of the planet is proportional to the permeability of space, the charge of the electron, the electron density of the surrounding medium, the angular velocity of rotation, and the square of the radius.
So, now he puts in the J from equation 5 into equation 2. Still this could maybe work if the planet is charged and if there is already a background magnetic field in the plasma in which the planet is embedded and I am still sure that there is a confusion of two B's which are confusingly not labeled.
Now, the magnetic moment M is defined as the total flux multiplied by the distance between the magnetic poles. Using a simple cylindrical model, as shown in Fig. 2, We find,
[latex]M=\mu_0en_e{\omega}R^2(2{\pi}R^3)=2{{\pi}\mu_0}en_e{\omega}R^5[/latex] (7)
This kind of definition for magnetic moment I have never seen.
Normally, the magnetic field of a magnetic moment m is given as:
B = (mu0 / 4 pi) (3 n(m.n - m) / r3
where n is a unit vector in direction r and m is the magnetic moment of a dipole (of which Alexeff is talking I think). Naturally, as Alexeff is not to thrilled about differentiation, integration and vectors, he can easily invert the above equation for the poles of the planet. Maybe the same comes out. Actually, mu0 seems to fall out then, typo by Zeuzzz or an error by Alexeff?
Next, consider the angular momentum L of a massM at radius R from the axis
L = MVR = MωR2. (8)
For a rotating cylindrical body, as shown in Fig. 2, I must integrate over the volume and assume a mass density m
[latex]L={\pi}m{\omega}R^5.[/latex] (9)
Sure why not, interesting detail, we are dealing with a cylinder??? Why? Is the angular momentum for a sphere too complicated to calculate? Anywho: L = MωR2 for a particle at R of mass M and angular velocity ω. For a sphere of constant density (why not, life is easy) we can quickly integrate L for the total angular momentum
Ltot = int int int MωR2 2R2 dR dphi dtheta,
where M and ω are constants (actually M is then a density but whatever) and doing the integration one finds that
Ltot = M ω (4 pi2/3) R5
Ah well, the same sortof.
Thus, the ratio of magnetic moment M to angular momentum L does not depend on R or ω
M/L = (2μ0ene)/m (10)
The remarkable result is that these simple calculations reproduce the correct proportionality ratio between magnetic moment L and angular momentum M over 12 orders of magnitude.
Well, indeed in this calculation R falls out of the ratio. However, this "calculation" is so riddled in unclarities from the start, that it is hard to know whether this has any significance on the graph or not.



