Looking At Electromagnetic Fields In The Solar System And Their Relation To Formation Of Magnetospheres

 

Magnetic and electric field are fundamental properties in the entire universe. Massive magnetic fields exist in the vicinity of pulsars, in active galactic nuclei, and probably around black holes. In the solar system most of the planets have an intrinsic magnetic field and in most cases the dipole component is the dominant contribution in particular to the far magnetic field. The table displayed at the top of this post gives the magnitude of each magnetic moment.

The Sun is an exception in that the dipole component is usually small compared to locally generated fields characteristic of the planets. The magnetic fields of most planets are caused by internal dynamo mechanisms in the outer core.

 These dynamos are driven by the thermal gradient between the inner core and the crust. Similarly the magnetic field on the Sun is generated in a turbulent convection zone below the photosphere. However, the turbulent convection in the sun generates a much more turbulent magnetic field such that the field in solar prominences is much stronger than the dipole component.

In a plasma the magnetic field is usually much stronger than the electric fields. The reason for this is the effective shielding of electric fields by the collective behavior of electric charges. Vice versa the magnetic field generated by electric currents is not shielded and therefore a long range force field. It is instructive  here to examine the properties of the electromagnetic field under Lorentz transformations of the form:

and:

                                                                             

with the Lorentz factor:   g _L     =  Ö(1 - v²  / c² )

 

 In a plasma the electric field is often caused by convection and such is of order:

O( v x B)

 

Fig. 1. Comparison of different magnetospheres.

Such an E- field magnitude yields a modification of order:

             g _L (v²  / c² )     << 1

in the magnetic field.

 The Lorentz invariants of the electromagnetic field are:

 B ²   -  (E²  / c² )  = const.

E ·  B = const.

_CB_ _ #;:=<?>A@

The invariants demonstrate if the magnetic (electric) energy density is larger in one reference frame then it is larger in all reference frames.

If there is an electric field component parallel to the magnetic field it exists in all reference frames. If the magnetic field is perpendicular to the electric field in one frame it is perpendicular in all frames.   The physics of magnetic fields includes a concept called the adiabatic invariant. The most important parameter for planetary magnetospheres is the magnetic moment   which is a constant of the motion, e.g.

m _m = m(v⊥) 2 / 2B = const.

And has associated with it a gyration energy: 

E _G =  m _m B = m/ 2 (E/B) 2

We have a relation between the velocities and magnetic fields:

 [v _^² /B_max]  =     [v _|| ² /B_z] =    const.  or  

[v _^² / v _|| ²  ]  =  B_z /B_max

where we take  B_z  = B_min

This leads to a simple model for a magnetic mirror system is shown below with M1 and M2 the magnetic “mirrors”  separated by a distance L .  In this case between the magnetic poles of a planet, e.g. Earth.)

If we are careful to change L slowly, then one finds:

v _|| L = const.

Now, assume M1 is stationary and M2 moves toward M1 at a velocity,  v _m  , then the incident velocity relative to the wall is:

[(v _||   +  v _m)   -   v _m ]

And:

D v _||   =  - [- (v _||   +  v _m)   -   v _m ]  + v _|    =   2 v _m

Thus, with each reflection, the velocity changes by 2 v _m and the number of reflections per second will be:

 v _||  / 2L 

Further:

dv _||  /dt  =   2 v _m  (v _||  / 2L)  =  v _||  /L (-dL/dt)

=    - v _||  /L (dL/dt)

So that:

d/dt (v _||  L)  = 0

 In what's called the "guiding center" approximation, any trapped particles (i.e. from the solar wind) will display a bounce period (T = 2 p  / Ω )  between the two mirror points, where Ω is the gyrofrequency.  Typical particles in Earth's magnetosphere take about one second to bounce from pole to pole.  

Relevant here is the 'pitch angle' (a ).  A particle with a shallow pitch angle such that:

sin (a )  >  Ö (B_min / B _max )

will be stopped by the magnetic bottle effect as it approaches the pole. Once stopped it can turn around and travel down the field line back to the other pole - until it is stopped there. This is called the 'bounce motion' of the particle. As noted above, for Earth the bounce period is about one second.

It is important here to point out the poles of the dipole are not coincident with the geographic north or south poles, which are defined at latitudes 90 deg N, and 90 deg S, respectively. One positional computation has put them at 79 deg N, 70W, and 79 deg S, 70 E, respectively.

Particles that don't meet the pitch angle condition given above plow into the atmosphere is said planet before reaching their mirror points. On encountering molecules in the atmosphere - like N2 or O2 - the molecules become ionized.  Then on re-combination of the ionized molecules that ionization energy is detected as light - generating the colors associated with the aurora, e.g.

Aurora Lights Up Skies Friday Night In The Wake of Most Energetic Solar Storm In 21 Years 

This will also apply to other planets, of course, e.g.

Detecting Auroral Activity on Jupiter 

And:

Hubble Captures Vivid Auroras in Jupiter’s Atmosphere - NASA Science

The magnetic flux density defined for the Earth is given as (Hargreaves, The Solar-Terrestrial Environment, Cambridge University Press, p. 150):

B(r,L) = M/r ³  [ 1 + 3 sin 2 L] ^1/2 

where M is the dipole moment (M ~ 10 ^25.9  G cm ³ , with G in gauss), r the geocentric radial distance and L the magnetic latitude (e.g. 70 deg for the locations given earlier).

Generically, the vector magnetic field at vector position x due to a magnetic dipole of vector dipole moment m at the origin, is given by:

B = {3(m*x)x - x }2/ ‖ x‖ ³ 

Meanwhile, the field components in the r and L directions are respectively (Hargreaves, ibid.)

B(r) = - 2M sin(L) / r³ 

B(L) = M cos (L) / r³ 

The characteristic time for variation of the Earth's magnetic field is given by:

dB/dt = B(r,L)/t  

Note that the dimensions of the planetary magnetospheres (Fig. 1) can be considered as rough approximations to the relative size of the associated magnetic bottles that trap solar particles.  

In the case of the pulsar magnetospheres, these are not governed by core dynamo action like the planets but rather by super relativistic electron (e-) and positron (e+) plasmas flowing in super strong magnetic fields and subject to strong gravitational fields. The plasma induces magnetospheric currents and space charges which distort the overall electromagnetic field.

As for the galactic magnetosphere, it arises in a similar dynamo action to what causes Earth's magnetosphere. But in the galactic case it is not molten iron in the core in motion but rather charged interstellar gas. This moves through space as part of the galaxy's overall rotation.  (Whenever charged particles move they create a magnetic field and potential magnetosphere.)

See Also:

How Jupiter's Moons - Io & Europa- Modulate Jupiter's Magnetosphere