|
|
| Line 1: |
Line 1: |
| [[Image:l shell global dipole.png|thumb|Plot showing field lines (which, in three dimensions would describe "shells") for [[L-shell|L-values]] 1.5, 2, 3, 4 and 5 using a dipole model of the Earth's magnetic field]]
| | I'm Faye and I live in a seaside city in northern United States, New York. I'm 38 and I'm will soon finish my study at Graduate School.<br><br>Also visit my web blog ... [http://www.wallpaperhdquality.com/profile/toliddell Biking for modern life mountain bike sizing.] |
| | |
| The '''dipole model of the Earth's magnetic field''' is a first order approximation of the rather complex true [[Earth's magnetic field]]. Due to effects of the [[interplanetary magnetic field]], and the [[solar wind]], the dipole model is particularly inaccurate at high [[L-shell]]s (e.g., above L=3), but may be a good approximation for lower L-shells. For more precise work, or for any work at higher L-shells, a more accurate model that incorporates solar effects, such as the [[Tsyganenko magnetic field model]], is recommended.
| |
| | |
| ==Equations==
| |
| The following equations describe the dipole magnetic field.<ref name="walt">{{cite book |last= Walt |first= Martin |title= [[Introduction to Geomagnetically Trapped Radiation]] |pages = 29–33 |publisher= [[Cambridge University Press]] |location = New York, NY |year= 1994 |isbn= 0-521-61611-5}}</ref>
| |
| | |
| First, define <math>B_0</math> as the mean value of the magnetic field at the magnetic equator on the Earth's surface. Typically <math>B_0=3.12\times10^{-5}\ \textrm{T}</math>.
| |
| | |
| Then, the radial and azimuthal fields can be described as
| |
| | |
| <math>B_r = -2B_0\left(\frac{R_E}{r}\right)^3\cos\theta</math>
| |
| | |
| <math>B_\theta = -B_0\left(\frac{R_E}{r}\right)^3\sin\theta</math>
| |
| | |
| <math>|B| = B_0\left(\frac{R_E}{r}\right)^3 \sqrt{1 + 3\cos^2\theta}</math>
| |
| | |
| where <math>R_E</math> is the mean [[Earth radius|radius of the Earth]] (approximately 6370 km), <math>r</math> is the radial distance from the center of the Earth (using the same units as used for <math>R_E</math>), and <math>\theta</math> is the azimuth measured from the north magnetic pole.
| |
| | |
| It is sometimes more convenient to express the magnetic field in terms of magnetic latitude and distance in earth radii. The magnetic latitude <math>\lambda</math> is measured northwards from the equator (analogous to [[latitude|geographic latitude]]) and is related to <math>\theta</math> by <math>\lambda = \pi/2 - \theta</math>. In this case, the radial and azimuthal components of the magnetic field (the latter still in the <math>\theta</math> direction, measured from the axis of the north pole) are given by
| |
| | |
| <math>B_r = -\frac{2B_0}{R^3}\sin\lambda</math>
| |
| | |
| <math>B_\theta = \frac{B_0}{R^3}\cos\lambda</math>
| |
| | |
| <math>|B| = \frac{B_0}{R^3} \sqrt{1 + 3\sin^2\lambda}</math>
| |
| | |
| where <math>R</math> in this case has units of Earth radii (<math>R = r/R_E</math>).
| |
| | |
| ==Invariant latitude==
| |
| Invariant latitude is a parameter that describes where a particular magnetic field line touches the surface of the Earth. It is given by<ref name="Kivelson-Russell">{{cite book |last1= Kivelson |first1= Margaret |last2= Russell |first2= Christopher |title= [[Introduction to Space Physics]] |pages = 166–167 |publisher= [[Cambridge University Press]] |location = New York, NY |year= 1995 |isbn= 0-521-45714-9}}</ref>
| |
| | |
| <math>\Lambda = \arccos\left(\sqrt{1/L}\right)</math>
| |
| | |
| or
| |
| | |
| <math>L = 1/\cos^2\left(\Lambda\right)</math>
| |
| | |
| where <math>\Lambda</math> is the invariant latitude and <math>L</math> is the L-shell describing the magnetic field line in question.
| |
| | |
| On the surface of the earth, the invariant latitude (<math>\Lambda</math>) is equal to the magnetic latitude (<math>\lambda</math>).
| |
| | |
| ==See also==
| |
| *[[Earth's magnetic field]]
| |
| *[[Dipole]]
| |
| *[[L-shell]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==External links==
| |
| *[http://ccmc.gsfc.nasa.gov/models/modelinfo.php?model=Tsyganenko%20Model Instant run of Tsyganenko magnetic field model] from NASA CCMC
| |
| *[http://geo.phys.spbu.ru/~tsyganenko/modeling.html Nikolai Tsyganenko's website] including Tsyganenko model source code
| |
| | |
| {{DEFAULTSORT:Dipole Model Of The Earth's Magnetic Field}}
| |
| [[Category:Geomagnetism]]
| |
| [[Category:Space physics]]
| |
I'm Faye and I live in a seaside city in northern United States, New York. I'm 38 and I'm will soon finish my study at Graduate School.
Also visit my web blog ... Biking for modern life mountain bike sizing.