A Model of Coronal A Model of Coronal Helmets with Prominences - - PowerPoint PPT Presentation

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 helmet’ ’s magnetic field, allowing magnetic s magnetic field, allowing magnetic 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

ˆ sin 1 sin sin 4 4

2 2 2

=

• +
• =
• A

r r A r c B c J r r r

• =
• =
• ˆ

sin ) , ( r r A B B

helmet

r r r r

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

x x x y x x y x y x x x y x y x x y x y

• +

=

• +
• +

=

• 2

) ( ) ( ) ( ) ( 2 ) (

2 2 2

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 5

1 ] [ 1

ssh r sun p

ds s B B GM M

µ

• 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 prominence comes out to be 6.6 e16 g 6.6 e16 g

Minimum Case Minimum Case

In order to just exceed the open field limit In order to just exceed the open field limit

• nly 5% of the Sun
• nly 5% of the Sun’

’s flux density need s flux density need 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 A “ “Normal Normal” ” Configuration Configuration

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

• f a coronal helmet
• f 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. Fong, B., B. C. Low, and Y. Fan. “ “Quiescent Quiescent 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. Low, B. C. “ “Models of Partially Open Models of Partially Open 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 Numerical Recipes in Fortran 77. 2nd ed. 1992. . 2nd ed. 1992.

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