A Model of Coronal A Model of Coronal Helmets with Prominences Helmets with Prominences Eric Greenfield Eric Greenfield Yuhong Fan Yuhong Fan B.C. Low B.C. Low High Altitude Observatory High Altitude Observatory
Outline Outline Background and Motivation Background and Motivation Methodology Methodology Results Results Future Work Future Work
Image courtesy of NCAR, HAO
Image courtesy of NCAR, HAO
My Task My Task Construct a simple magnetic model of a coronal Construct a simple magnetic model of a coronal helmet, capturing the basic characteristics of the helmet, capturing the basic characteristics of the partially open configuration partially open configuration Investigate how the shape of the helmet Investigate how the shape of the helmet depends on surface flux distribution depends on surface flux distribution Investigate the possibility that a prominence Investigate the possibility that a prominence sheet within a helmet cavity can anchor the sheet within a helmet cavity can anchor the helmet’ ’s magnetic field, allowing magnetic s magnetic field, allowing magnetic helmet energy to build beyond the open field limit energy to build beyond the open field limit Can this mechanism store enough energy for Can this mechanism store enough energy for driving a CME? driving a CME?
Outline Outline Background and Motivation Background and Motivation Methodology Methodology Results Results Future Work Future Work
Magnetic Field Constraints Magnetic Field Constraints The magnetic field of a coronal helmet must The magnetic field of a coronal helmet must satisfy certain conditions satisfy certain conditions r r r r A ( r , ) � � � ˆ B 0 B � • = � = � � � � � helmet r sin � � � r r r 2 c c A sin 1 A � � � � � � � � ˆ J B 0 � � = � � � � + � = � � � � 2 2 4 4 r sin r r sin � � � � � � � � � � � � �
New Coordinate System New Coordinate System
Discretization and a Numerical Discretization and a Numerical Solution Solution The equation must be discretized to solve The equation must be discretized to solve numerically numerically 2 y y ( x x ) 2 y ( x ) y ( x x ) � + � � + � � = 2 2 x x � � y y ( x x ) y ( x x ) � + � � � � = x 2 x � �
Outline Outline Background and Motivation Background and Motivation Methodology Methodology Results Results Future Work Future Work
Same Problem, Different Solutions Same Problem, Different Solutions
Open vs. Partially Open vs. Closed Open vs. Partially Open vs. Closed
Changing the Shape of the Helmet Changing the Shape of the Helmet Additional flux concentrated near the Additional flux concentrated near the equator equator
Introducing a Prominence Introducing a Prominence In order to ensure that the helmet can In order to ensure that the helmet can store enough magnetic energy to surpass store enough magnetic energy to surpass the open field limit, a prominence must be the open field limit, a prominence must be introduced to the model to act as an introduced to the model to act as an anchor anchor This is done by applying an additional This is done by applying an additional boundary condition as a current sheet boundary condition as a current sheet
New Boundary Condition New Boundary Condition
Helmet with Prominence Helmet with Prominence After adding the prominence, a closed system of After adding the prominence, a closed system of field lines is now present in the helmet cavity field lines is now present in the helmet cavity
Energy and Prominence Mass Energy and Prominence Mass In one case we set the amount of flux In one case we set the amount of flux passing through the sheet to be equal to passing through the sheet to be equal to 15% that of the entire Sun 15% that of the entire Sun The resulting ratio The resulting ratio E/Eopen = 1.86 E/Eopen = 1.86 is well is well beyond the open field limit beyond the open field limit
Prominence Mass Prominence Mass 1 1 1 M [ B B ] ds = � p r 0 � µ = 5 GM s sun ssh For the case of E/Eopen=1.86, the mass of the For the case of E/Eopen=1.86, the mass of the prominence comes out to be 6.6 e16 g 6.6 e16 g prominence comes out to be
Minimum Case Minimum Case In order to just exceed the open field limit In order to just exceed the open field limit only 5% of the Sun’ ’s flux density need s flux density need only 5% of the Sun thread through the prominence thread through the prominence E/Eopen = 1.003 E/Eopen = 1.003 Mp = 1.31 e16 g Mp = 1.31 e16 g
Exceeding the Open Field Limit Exceeding the Open Field Limit
A “ “Normal Normal” ” Configuration Configuration A Mp = 3.306 e16 g Mp = 3.306 e16 g
Outline Outline Background and Motivation Background and Motivation Methodology Methodology Results Results Future Work Future Work
Future Work Future Work This model examines only a very idealized case This model examines only a very idealized case of a coronal helmet of a coronal helmet The model can be improved by combining the The model can be improved by combining the prominence with a varying surface flux function prominence with a varying surface flux function Construct a prominence with a normal rather Construct a prominence with a normal rather than inverse configuration than inverse configuration The helmet can also be moved away from the The helmet can also be moved away from the equator equator The current sheet at the top of the helmet must still go The current sheet at the top of the helmet must still go to the equator to the equator This would mean solving a boundary value problem This would mean solving a boundary value problem with the curved shape of the current as an unknown with the curved shape of the current as an unknown
Resources Resources Fong, B., B. C. Low, and Y. Fan. “ “Quiescent Quiescent Fong, B., B. C. Low, and Y. Fan. Solar Prominences and Magnetic-Energy Solar Prominences and Magnetic-Energy Storage.” Storage. ” Low, B. C., B. Fong, and Y. Fan. "The Mass of a Low, B. C., B. Fong, and Y. Fan. "The Mass of a Solar Quiescent Prominence.“ “ Solar Quiescent Prominence. Low, B. C. “ “Models of Partially Open Models of Partially Open Low, B. C. Magnetospheres with and without Magnetodisks” ” Magnetospheres with and without Magnetodisks William, Press H., and Brian P. Flannery. William, Press H., and Brian P. Flannery. Numerical Recipes in Fortran 77. 2nd ed. 1992. Numerical Recipes in Fortran 77 . 2nd ed. 1992.
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