Galactic Magnetic Fields & Rotation Curves

[Widdekind]

REVISED UPDATE

 

(1) Circumstellar Disk Analogy — Magnetic Field Lines transfer Angular Momentum outwards

 

According to Jayant V. Narlikar (From Black Clouds to Black Holes, pp. 61-64), magnetic field-lines, threading through circumstellar disks, can spin up outer disk regions, while spinning down inner disk regions:

In Figure 5.9, a magnetic pole at A will be forced by the magnetic field in the cloud to move to B along the line of force. Now, these lines of

force have a tendency to stick on to cloud particles all the time. So, as A & B rotate about a common axis, the line of force moves w/ them. But, since A & B do not rotate at the same rate, w/ A rotating faster than B, the line of force gets twisted. And, when twisted, it tends to straighten itself.

 

In this process, it pulls A back, while making B go faster, this reversing the tendency of the cloud as a whole to rotate faster in the center & slower in the outskirts. This resistance offered by the line of force serves the two required needs: 1) it slows the central part down, 2) it pushes the outer parts further out by making them rotate faster.

 

 

 

 

(2) Similarly structured Magnetic Fields, in Galactic Disks, strongly influence dynamics

 

According to Carroll & Ostlie (Introduction to Modern Astrophysics, pg. 935), Galactic magnetic fields resemble those depicted above, for forming star systems:

Within the [Galactic] disk,

the field tends to follow the Galaxy' spiral arms

, and has a typical strength of 4 [math]\mu G[/math]... Although the global Galactic magnetic field is quite weak, relative to terrestrial fields, it likely plays a significant role in the structure & evolution of the Milky Way. This can be seen by considering its energy density, which appears to be comparable (perhaps equal) to the thermal energy density of gas within the disk.

 

 

 

(3) Estimation of Magnitude of Magnetic Tension Force in Galactic Disk

 

The Magnetic Tension Force:

[math]\frac{1}{\mu}\left( \vec{B} \cdot \vec{\nabla} \right) \vec{B} = \frac{1}{\mu} \; B \frac{d}{ds}[/math]

can be decomposed, into the component tangential to the Magnetic Field Lines, and the component normal to them:

 

[math]\vec{B_{t}} = \frac{\partial}{\partial S} \left (\frac{B^{2}}{2 \; \mu} \right) \hat{t}[/math]

[math]\vec{B_{n}} = \frac{B^{2}}{\mu \; R_{c}} \hat{n}[/math]

where R_c is the Radius of Curvature of the Magnetic Field Line, [math]\hat{t}[/math] is the outward-pointing unit tangent vector, [math]\hat{n}[/math] is the inward-pointing unit normal vector, and the derivative [math]\frac{\partial}{\partial S}[/math] is taken along the field line, in the direction of [math]\hat{t}[/math]*.

*

http://www-solar.mcs.st-and.ac.uk/~alan/sun_course/Chapter2/node15.html#fig13

Now, to lowest order, we may estimate the tangential derivative by using a Single Zone Approximation:

[math]\frac{\partial}{\partial S} \left (\frac{B^{2}}{2 \; \mu} \right) \approx \frac{\Delta (B^{2})}{2 \; \mu \; S} = \frac{B^{2} - B_{0}^{2}}{2 \; \mu \; S}[/math]

where S is the integrated Arc Length from the pole to the point in question, B₀ is the magnetic field strength at the pole, and B is the field strength at the point in question. This is depicted below. Note that, b/c the field strength decreases along Galactic field lines [math]\left( \frac{\partial}{\partial S} < 0 \right)[/math], the tangential component of the Magnetic Tension Force actually points along the inward-pointing tangent vector. To make a mechanical analogy, this is akin to keeping an axe-head on an axe-handle by virtue of the flaring, of the latter, outwards towards its tip:

 

 

Now, Spiral Galaxy Arms typically take the shape of a Logarithmic Spiral, with Pitch Angles typically between 10-40 degrees. For the Milky Way, that angle is about 12 degrees*. And, for Logarithmic Spirals, we have the following relations for the Radius of Curvature and the integrated Arc Length**:

[math]R_{c} = R \; \sqrt{1 + b^{2}}[/math]

 

[math]S = R \frac{\sqrt{1 + b^{2}}}{b}[/math]

where b is the reciprocal, of the tangent, of the compliment, of the Pitch Angle***.

*

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

**

http://www.2dcurves.com/spiral/spirallo.html

***

http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/KURSATgeometrypro/golden%20spiral/logspiral-history.html

Thus, we wish to compare the total (radial) force, acting on a parcel of fluid in the ISM [math]\left( \rho \frac{V^{2}}{R} \right)[/math], with the (radial) force from Magnetic Tension. For sake of simplicity, we shall presume a pretty plausible Pitch Angle of 45 degrees, for which case, b = 1, R_c = S = 2^1/2 R, and [math]\hat{-t} \cdot (-\hat{r}) = \hat{n} \cdot (-\hat{r}) = 2^{-1/2}[/math]. Then, the estimated inward-radial component, of the Magnetic Tension Force, is:

[math]\vec{F_{MT}} \cdot (-\hat{r}) = \left( \vec{B_{t}} + \vec{B_{n}} \right) \cdot (-\hat{r})[/math]

 

[math] = \frac{\partial}{\partial S} \left (\frac{B^{2}}{2 \; \mu} \right) \left( \hat{t} \cdot (-\hat{r}) \right) + \frac{B^{2}}{\mu \; R_{c}} \left( \hat{n} \cdot (-\hat{r}) \right)[/math]

 

[math]\approx \frac{B_{0}^{2} - B^{2}}{2 \; \mu \; 2^{1/2} \; R} 2^{1/2} + \frac{B^{2}}{\mu \; 2^{1/2} \; R} 2^{1/2}[/math]

 

[math] = \frac{B_{0}^{2} + B^{2}}{2 \; \mu \; R}[/math]

 

So, we define the ratio:

[math]\beta \equiv \frac{ \frac{B_{0}^{2} + B^{2}}{ 2 \; \mu \; R} }{\rho \frac{V^{2}}{R}} = \frac{1}{2 \; \mu \; \rho}\frac{B_{0}^{2} + B^{2}}{V^{2}}[/math]

For the Sun, and using values for the Warm Ionized Medium, we have that:

[math]B_{0} = 40 \; \mu G[/math]

 

[math]B = 4 \; \mu G[/math]

 

[math]V = 220 \; km \; s^{-1}[/math]

 

[math]\rho = (0.4 - 1) \times 8.4 \times 10^{-25} \; g \; cm^{-3}[/math]

so that:

[math]\beta \approx 0.16 - 0.39[/math]

And, these values could be up to 3 orders of magnitude greater, for the Hot Ionized Medium, which "makes up most of the ISM"*. Thus, Magnetic Tension could, quite conceivably, play an important (perhaps pivotal ?) role, in Spiral Galaxy Disk dynamics.

*

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

 

 

 

(4) Predicted correlation, between Galactic Magnetic Field Strengths, and Spiral Arm Pitch Angles

 

If Galactic magnetic fields dominate disk dynamics, and if said fields thread through spiral arms, then since magnetic tension tries to straighten field lines, therefore Galaxies possessing more powerful magnetic fields should show straighter spiral arms, having higher Pitch Angles.

Edited September 3, 2009 by Widdekind
Consecutive posts merged.