Calculating Magnetic Energy in Active Regions Using Coronal Loopsfor AR 11092 and AR 11112
Presentation by,Joseph B. Jensen
Magnetic energy equation
∇ ×Β =α ( x ) Β ( x )
W =1 /8 π ∫∣Β 2∣d 3 x=1 /8 π ∫d Φ ∫ Β dl
Non-linear force-free equilibrium for coronal field
Force = J × Β + ∇ p + ρ g = 0
∫ab sin ( x) ≈ (b−a )/N ∑
i=1
N
sin ( xi )
Monte Carlo Method
∫ab sin ( x) ≈ (b−a )/N ∑
i=1
N
sin ( xi )/ p (x i )
W N = 1 /N ∑i=1
N
∣Β z (x i , y i , 0 )∣C ( x i , y i )/8 π p ( x i , y i )
where
C ( x , y )=∫L ( x , y )B [ x( l )]dl
Calculating Total Magnetic Energy
Φ i =∣Β z ( x i , y i , 0)∣/ Np ( x i , y i )and
= ∑i=1
N
Φ iC ( x i , y i)/8 π
Calculating the Free Energy
W free= 1 /8π ∫d Φ ∫(Β−ΒP )dl ≈ ∑i=1
N
ΦiΔ C i /8 π
Δ C i ( x , y ) = ∫L i (Β−Β P ) •dl
where
Β total − Β free = Β potential
Ar11092 at 0500 UT
Ar11092 at 1700 UT
7.2 x 1032 ergs − 2.6 x 1032ergs=4.6 x 10 32ergs
4.5 x 1032 ergs
4.7 x 10 32ergs − 1.0 x 10 32ergs = 3.7 x 1032 ergs
3.8 x 1032ergs
Ar11112 at 1650 UT
8.5 x 1031ergs − 1.1 x 1031ergs = 7.4 x 10 31ergs
8.7 x 1031ergs
Β total − Β free = Β potential
Lower bound on the free energy of
8.25 x 1030ergs
Conclusion
-The four different wavelengths agree well with each other.
-This method is in good agreement with some other methods of energy calculation.
References
Special Thanks to
Dr. Dana Longcope and
Lucas Tarr for programming help
Malanushenko, A., Longcope, D. W., & McKenzie, D. E. “ Reconstructing the Local Twist of Coronal Magnetic Fields and the Three-Dimensional Shape of the Field Lines from Coronal Loops in EUV and X-Ray Images” 2009, ApJ, 707, 1044
Longcope, D. W., Mulanushenko, A., “Computing Magnetic Energy from Coronal Imaging Data” 2012, journal of Still Working On It, 24, 456
Tarr, L., Longcope,D., “Calculating Energy Storage Due To Topological Changes In Emerging Active Region NOAA AR 11112” 2012, ApJ, 749:64
Monte Carlo method. (2012, August 6). In Wikipedia, The Free Encyclopedia. Retrieved 16:24, August 6, 2012, from http://en.wikipedia.org/w/index.php