Mohammed Humed Yusuf



                               




                                                                    

                   BEREA COLLEGE                                                                                                                                                                                

Solar Physics REU program-Summer 2009
Montana State University
Bozeman, MT 59717-3840
yusufm@berea.edu



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Personal Information

Welcome to my homepage! I am Humed and I hail from Dhaka, capital of Bangladesh. I came to the United States in the fall of 2006 to pursue my undergraduate degree in Physics and Mathematics at Berea College, where I will be a rising senior in the upcoming fall semester. During the summer of 2007, I was involved in undergraduate research at Berea College with Professor Tracy Hodge doing differential photometry with our 16" Ritchey-Cretien telescope. For the summer of 2008, I had the privilege of attending Harvard University for summer classes made possible through a scholarship (Ballard McConnell Willis Award) from the Berea College Mathematics Department.

This summer I am doing a research project at MSU basically  trying out a novel algorithm of constructing magnetic field lines from the TRACE satellite images taken some time around the year 2000. The algorithm was developed by my mentor Anna Malanushenko under the guidance of Dr. Dana Longcope. The algorithm is based on the idea of a force-free magnetic field model, where it is assumed that magnetic field loops emanating from the sunspots onto the corona remain in equilibrium for short periods of time. More about this is discussed lower in the page.

Preliminary Topics - An introduction to force free magnetic field


The theory

If the forces acting on a coronal magnetic field are taken into account, then the prevalence of three forces are accounted for. These forces are namely the force due to gravity, the force due to the pressure of the underlying plasma and the Lorentz force. The Lorentz force, by definition, is a combination of electric force and magnetic force. It is in fact the force experienced by charged particles in electromagnetic fields. While considering the forces in coronal magnetic fields, it has been shown that Lorentz force overshadows all the other forces. Moreover, since "force-free" literally means that the net force  is zero, the coronal magnetic fields can be assumed to be somewhat in a state of equilibrium. If the net force is zero and Lorentz force is the dominant force, then it means that the Lorentz force is zero in the force free magnetic field model. Shown below is a quick and dirty derivation of the condition underlying the force free magnetic field model.
      
Starting with the Lorentz Force:   

                                                                                                                       F=(qv) xB + qE  .......(1)
     
Plasma is mainly ionized gas and is a conductor. It is known for a fact that electric field inside a conductor is zero. Also, inside the plasma, the charge densities of the positive ions and the electrons are approximately same resulting in a zero net charge. So, the term qE is zero and the Lorentz force can be written as:

                                                                                                                       F=(qv) xB  ......(2)

But,

                                                                                                                       Current = J = qv  .......(3)

As the Lorentz force is zero, the following relation holds for  J(r) and B(r)

                                                                                                                       J(r) x B(r) = 0   .......(4)

Since J(r) x B(r) = 0,  J(r)  and B(r) are parallel. So,
      
                                                                                                                      J(r) = α (r) .B(r)  .......(5)
 
   
One of Maxwell's equations is:  

                                                                                                                      ∇XB = (1/c) . (∂E/∂t)+ 4πJ  .......(6)
  
For E = 0, the equation becomes
              
                                                                                                                       ∇XB = 4πJ  .......(7)
    
By equating  (5) and (7), the following equation is obtained

                                                                                                                       ∇XB = α (r) .B(r) .......(8)
  
Whenever solar physicists say force-free magnetic field, they refer to the above equation which was derived by assuming the Helmholtz force to be zero in the coronal region.


The space dependent scalar , α

The scalar α is important in determining the topology of the coronal magnetic field. Mathematically, it can be shown that α determines the twistedness of any magnetic field prevailing between two dipoles, which are sunspots in case of the sun. In fact, it can be shown that  α =4πn, where n is the number of field lines per unit length. Now, several values of α have been tried into equation (8) to best model the physical realities of the coronal magnetic fields. If α=0, then it gives us the uninteresting case of ∇XB = 0. From the basics of electromagnetic theory it can then be said that B = ∇Φ, which eventually turns into Laplace's Equation, ∇^2Φ =0, by substituting into one of Maxwell's equations. The Laplace's Equation is a partial differential equation that can be solved analytically if the boundary conditions are known and hence making α=0 an uninteresting case. The situation gets more interesting when α is something other than zero. In that case, the partial differential equation (8) has to be solved numerically to get a proper representation of the magnetic field in the corona.

Motivation


Purpose of study

The loops that are visible in extreme untra-violet images of the sun are our subjects of study. What are these loops? Some hypothesize that they are just loops of current, others say that they are actually loops of magnetic field lines. Whatever they are, they form interesting subjects of study and scientific research. The stand we would be taking is an attempt to support that  they are indeed magnetic field lines. And our model would be the force free magnetic field model. Using this as a tool, our goal would be to model three dimesional magnetic field lines. For plotting these field lines we need to find the values of α and z that would best represent the loops. Why z? Finding z is crucial because when we see any image, we cannot perceive how high the loops are. So our best bet would be to use a program that would take the x and y coordiantes of the loops and go through a range of values of z and α  giving us the best possible magnetic field line for a given loop. In real life scenario, constant α fields are an improbability. Different α s are needed for different regions of coronal magnetic fields. However, the constant α field model provides us with an important tool to represent complicated magnetic fields. One would assume that this model would not work, but surprisingly it does.



Photo credit: Getty Images

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